src/HOL/UNITY/Extend.thy
 author wenzelm Thu Jul 23 22:13:42 2015 +0200 (2015-07-23) changeset 60773 d09c66a0ea10 parent 58889 5b7a9633cfa8 child 61424 c3658c18b7bc permissions -rw-r--r--
more symbols by default, without xsymbols mode;
1 (*  Title:      HOL/UNITY/Extend.thy
2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1998  University of Cambridge
5 Extending of state setsExtending of state sets
6   function f (forget)    maps the extended state to the original state
7   function g (forgotten) maps the extended state to the "extending part"
8 *)
10 section{*Extending State Sets*}
12 theory Extend imports Guar begin
14 definition
15   (*MOVE to Relation.thy?*)
16   Restrict :: "[ 'a set, ('a*'b) set] => ('a*'b) set"
17   where "Restrict A r = r \<inter> (A <*> UNIV)"
19 definition
20   good_map :: "['a*'b => 'c] => bool"
21   where "good_map h <-> surj h & (\<forall>x y. fst (inv h (h (x,y))) = x)"
22      (*Using the locale constant "f", this is  f (h (x,y))) = x*)
24 definition
25   extend_set :: "['a*'b => 'c, 'a set] => 'c set"
26   where "extend_set h A = h ` (A <*> UNIV)"
28 definition
29   project_set :: "['a*'b => 'c, 'c set] => 'a set"
30   where "project_set h C = {x. \<exists>y. h(x,y) \<in> C}"
32 definition
33   extend_act :: "['a*'b => 'c, ('a*'a) set] => ('c*'c) set"
34   where "extend_act h = (%act. \<Union>(s,s') \<in> act. \<Union>y. {(h(s,y), h(s',y))})"
36 definition
37   project_act :: "['a*'b => 'c, ('c*'c) set] => ('a*'a) set"
38   where "project_act h act = {(x,x'). \<exists>y y'. (h(x,y), h(x',y')) \<in> act}"
40 definition
41   extend :: "['a*'b => 'c, 'a program] => 'c program"
42   where "extend h F = mk_program (extend_set h (Init F),
43                                extend_act h ` Acts F,
44                                project_act h -` AllowedActs F)"
46 definition
47   (*Argument C allows weak safety laws to be projected*)
48   project :: "['a*'b => 'c, 'c set, 'c program] => 'a program"
49   where "project h C F =
50        mk_program (project_set h (Init F),
51                    project_act h ` Restrict C ` Acts F,
52                    {act. Restrict (project_set h C) act :
53                          project_act h ` Restrict C ` AllowedActs F})"
55 locale Extend =
56   fixes f     :: "'c => 'a"
57     and g     :: "'c => 'b"
58     and h     :: "'a*'b => 'c"    (*isomorphism between 'a * 'b and 'c *)
59     and slice :: "['c set, 'b] => 'a set"
60   assumes
61     good_h:  "good_map h"
62   defines f_def: "f z == fst (inv h z)"
63       and g_def: "g z == snd (inv h z)"
64       and slice_def: "slice Z y == {x. h(x,y) \<in> Z}"
67 (** These we prove OUTSIDE the locale. **)
70 subsection{*Restrict*}
71 (*MOVE to Relation.thy?*)
73 lemma Restrict_iff [iff]: "((x,y): Restrict A r) = ((x,y): r & x \<in> A)"
74 by (unfold Restrict_def, blast)
76 lemma Restrict_UNIV [simp]: "Restrict UNIV = id"
77 apply (rule ext)
78 apply (auto simp add: Restrict_def)
79 done
81 lemma Restrict_empty [simp]: "Restrict {} r = {}"
82 by (auto simp add: Restrict_def)
84 lemma Restrict_Int [simp]: "Restrict A (Restrict B r) = Restrict (A \<inter> B) r"
85 by (unfold Restrict_def, blast)
87 lemma Restrict_triv: "Domain r \<subseteq> A ==> Restrict A r = r"
88 by (unfold Restrict_def, auto)
90 lemma Restrict_subset: "Restrict A r \<subseteq> r"
91 by (unfold Restrict_def, auto)
93 lemma Restrict_eq_mono:
94      "[| A \<subseteq> B;  Restrict B r = Restrict B s |]
95       ==> Restrict A r = Restrict A s"
96 by (unfold Restrict_def, blast)
98 lemma Restrict_imageI:
99      "[| s \<in> RR;  Restrict A r = Restrict A s |]
100       ==> Restrict A r \<in> Restrict A ` RR"
101 by (unfold Restrict_def image_def, auto)
103 lemma Domain_Restrict [simp]: "Domain (Restrict A r) = A \<inter> Domain r"
104 by blast
106 lemma Image_Restrict [simp]: "(Restrict A r) `` B = r `` (A \<inter> B)"
107 by blast
109 (*Possibly easier than reasoning about "inv h"*)
110 lemma good_mapI:
111      assumes surj_h: "surj h"
112          and prem:   "!! x x' y y'. h(x,y) = h(x',y') ==> x=x'"
113      shows "good_map h"
114 apply (simp add: good_map_def)
115 apply (safe intro!: surj_h)
116 apply (rule prem)
117 apply (subst surjective_pairing [symmetric])
118 apply (subst surj_h [THEN surj_f_inv_f])
119 apply (rule refl)
120 done
122 lemma good_map_is_surj: "good_map h ==> surj h"
123 by (unfold good_map_def, auto)
125 (*A convenient way of finding a closed form for inv h*)
126 lemma fst_inv_equalityI:
127      assumes surj_h: "surj h"
128          and prem:   "!! x y. g (h(x,y)) = x"
129      shows "fst (inv h z) = g z"
130 by (metis UNIV_I f_inv_into_f pair_collapse prem surj_h)
133 subsection{*Trivial properties of f, g, h*}
135 context Extend
136 begin
138 lemma f_h_eq [simp]: "f(h(x,y)) = x"
139 by (simp add: f_def good_h [unfolded good_map_def, THEN conjunct2])
141 lemma h_inject1 [dest]: "h(x,y) = h(x',y') ==> x=x'"
142 apply (drule_tac f = f in arg_cong)
143 apply (simp add: f_def good_h [unfolded good_map_def, THEN conjunct2])
144 done
146 lemma h_f_g_equiv: "h(f z, g z) == z"
147 by (simp add: f_def g_def
148             good_h [unfolded good_map_def, THEN conjunct1, THEN surj_f_inv_f])
150 lemma h_f_g_eq: "h(f z, g z) = z"
151 by (simp add: h_f_g_equiv)
154 lemma split_extended_all:
155      "(!!z. PROP P z) == (!!u y. PROP P (h (u, y)))"
156 proof
157    assume allP: "\<And>z. PROP P z"
158    fix u y
159    show "PROP P (h (u, y))" by (rule allP)
160  next
161    assume allPh: "\<And>u y. PROP P (h(u,y))"
162    fix z
163    have Phfgz: "PROP P (h (f z, g z))" by (rule allPh)
164    show "PROP P z" by (rule Phfgz [unfolded h_f_g_equiv])
165 qed
167 end
170 subsection{*@{term extend_set}: basic properties*}
172 lemma project_set_iff [iff]:
173      "(x \<in> project_set h C) = (\<exists>y. h(x,y) \<in> C)"
174 by (simp add: project_set_def)
176 lemma extend_set_mono: "A \<subseteq> B ==> extend_set h A \<subseteq> extend_set h B"
177 by (unfold extend_set_def, blast)
179 context Extend
180 begin
182 lemma mem_extend_set_iff [iff]: "z \<in> extend_set h A = (f z \<in> A)"
183 apply (unfold extend_set_def)
184 apply (force intro: h_f_g_eq [symmetric])
185 done
187 lemma extend_set_strict_mono [iff]:
188      "(extend_set h A \<subseteq> extend_set h B) = (A \<subseteq> B)"
189 by (unfold extend_set_def, force)
191 lemma (in -) extend_set_empty [simp]: "extend_set h {} = {}"
192 by (unfold extend_set_def, auto)
194 lemma extend_set_eq_Collect: "extend_set h {s. P s} = {s. P(f s)}"
195 by auto
197 lemma extend_set_sing: "extend_set h {x} = {s. f s = x}"
198 by auto
200 lemma extend_set_inverse [simp]: "project_set h (extend_set h C) = C"
201 by (unfold extend_set_def, auto)
203 lemma extend_set_project_set: "C \<subseteq> extend_set h (project_set h C)"
204 apply (unfold extend_set_def)
205 apply (auto simp add: split_extended_all, blast)
206 done
208 lemma inj_extend_set: "inj (extend_set h)"
209 apply (rule inj_on_inverseI)
210 apply (rule extend_set_inverse)
211 done
213 lemma extend_set_UNIV_eq [simp]: "extend_set h UNIV = UNIV"
214 apply (unfold extend_set_def)
215 apply (auto simp add: split_extended_all)
216 done
218 subsection{*@{term project_set}: basic properties*}
220 (*project_set is simply image!*)
221 lemma project_set_eq: "project_set h C = f ` C"
222 by (auto intro: f_h_eq [symmetric] simp add: split_extended_all)
224 (*Converse appears to fail*)
225 lemma project_set_I: "!!z. z \<in> C ==> f z \<in> project_set h C"
226 by (auto simp add: split_extended_all)
229 subsection{*More laws*}
231 (*Because A and B could differ on the "other" part of the state,
232    cannot generalize to
233       project_set h (A \<inter> B) = project_set h A \<inter> project_set h B
234 *)
235 lemma project_set_extend_set_Int: "project_set h ((extend_set h A) \<inter> B) = A \<inter> (project_set h B)"
236   by auto
238 (*Unused, but interesting?*)
239 lemma project_set_extend_set_Un: "project_set h ((extend_set h A) \<union> B) = A \<union> (project_set h B)"
240   by auto
242 lemma (in -) project_set_Int_subset:
243     "project_set h (A \<inter> B) \<subseteq> (project_set h A) \<inter> (project_set h B)"
244   by auto
246 lemma extend_set_Un_distrib: "extend_set h (A \<union> B) = extend_set h A \<union> extend_set h B"
247   by auto
249 lemma extend_set_Int_distrib: "extend_set h (A \<inter> B) = extend_set h A \<inter> extend_set h B"
250   by auto
252 lemma extend_set_INT_distrib: "extend_set h (INTER A B) = (\<Inter>x \<in> A. extend_set h (B x))"
253   by auto
255 lemma extend_set_Diff_distrib: "extend_set h (A - B) = extend_set h A - extend_set h B"
256   by auto
258 lemma extend_set_Union: "extend_set h (Union A) = (\<Union>X \<in> A. extend_set h X)"
259   by blast
261 lemma extend_set_subset_Compl_eq: "(extend_set h A \<subseteq> - extend_set h B) = (A \<subseteq> - B)"
262   by (auto simp: extend_set_def)
265 subsection{*@{term extend_act}*}
267 (*Can't strengthen it to
268   ((h(s,y), h(s',y')) \<in> extend_act h act) = ((s, s') \<in> act & y=y')
269   because h doesn't have to be injective in the 2nd argument*)
270 lemma mem_extend_act_iff [iff]: "((h(s,y), h(s',y)) \<in> extend_act h act) = ((s, s') \<in> act)"
271   by (auto simp: extend_act_def)
273 (*Converse fails: (z,z') would include actions that changed the g-part*)
274 lemma extend_act_D: "(z, z') \<in> extend_act h act ==> (f z, f z') \<in> act"
275   by (auto simp: extend_act_def)
277 lemma extend_act_inverse [simp]: "project_act h (extend_act h act) = act"
278   unfolding extend_act_def project_act_def by blast
280 lemma project_act_extend_act_restrict [simp]:
281      "project_act h (Restrict C (extend_act h act)) =
282       Restrict (project_set h C) act"
283   unfolding extend_act_def project_act_def by blast
285 lemma subset_extend_act_D: "act' \<subseteq> extend_act h act ==> project_act h act' \<subseteq> act"
286   unfolding extend_act_def project_act_def by force
288 lemma inj_extend_act: "inj (extend_act h)"
289 apply (rule inj_on_inverseI)
290 apply (rule extend_act_inverse)
291 done
293 lemma extend_act_Image [simp]:
294      "extend_act h act `` (extend_set h A) = extend_set h (act `` A)"
295   unfolding extend_set_def extend_act_def by force
297 lemma extend_act_strict_mono [iff]:
298      "(extend_act h act' \<subseteq> extend_act h act) = (act'<=act)"
299   by (auto simp: extend_act_def)
301 lemma [iff]: "(extend_act h act = extend_act h act') = (act = act')"
302   by (rule inj_extend_act [THEN inj_eq])
304 lemma (in -) Domain_extend_act:
305     "Domain (extend_act h act) = extend_set h (Domain act)"
306   unfolding extend_set_def extend_act_def by force
308 lemma extend_act_Id [simp]: "extend_act h Id = Id"
309   unfolding extend_act_def by (force intro: h_f_g_eq [symmetric])
311 lemma project_act_I:  "!!z z'. (z, z') \<in> act ==> (f z, f z') \<in> project_act h act"
312   unfolding project_act_def by (force simp add: split_extended_all)
314 lemma project_act_Id [simp]: "project_act h Id = Id"
315   unfolding project_act_def by force
317 lemma Domain_project_act: "Domain (project_act h act) = project_set h (Domain act)"
318   unfolding project_act_def by (force simp add: split_extended_all)
321 subsection{*extend*}
323 text{*Basic properties*}
325 lemma (in -) Init_extend [simp]:
326      "Init (extend h F) = extend_set h (Init F)"
327   by (auto simp: extend_def)
329 lemma (in -) Init_project [simp]:
330      "Init (project h C F) = project_set h (Init F)"
331   by (auto simp: project_def)
333 lemma Acts_extend [simp]: "Acts (extend h F) = (extend_act h ` Acts F)"
334   by (simp add: extend_def insert_Id_image_Acts)
336 lemma AllowedActs_extend [simp]:
337      "AllowedActs (extend h F) = project_act h -` AllowedActs F"
338   by (simp add: extend_def insert_absorb)
340 lemma (in -) Acts_project [simp]:
341      "Acts(project h C F) = insert Id (project_act h ` Restrict C ` Acts F)"
342   by (auto simp add: project_def image_iff)
344 lemma AllowedActs_project [simp]:
345      "AllowedActs(project h C F) =
346         {act. Restrict (project_set h C) act
347                \<in> project_act h ` Restrict C ` AllowedActs F}"
348 apply (simp (no_asm) add: project_def image_iff)
349 apply (subst insert_absorb)
350 apply (auto intro!: bexI [of _ Id] simp add: project_act_def)
351 done
353 lemma Allowed_extend: "Allowed (extend h F) = project h UNIV -` Allowed F"
354   by (auto simp add: Allowed_def)
356 lemma extend_SKIP [simp]: "extend h SKIP = SKIP"
357 apply (unfold SKIP_def)
358 apply (rule program_equalityI, auto)
359 done
361 lemma (in -) project_set_UNIV [simp]: "project_set h UNIV = UNIV"
362   by auto
364 lemma (in -) project_set_Union: "project_set h (Union A) = (\<Union>X \<in> A. project_set h X)"
365   by blast
368 (*Converse FAILS: the extended state contributing to project_set h C
369   may not coincide with the one contributing to project_act h act*)
370 lemma (in -) project_act_Restrict_subset:
371      "project_act h (Restrict C act) \<subseteq> Restrict (project_set h C) (project_act h act)"
372   by (auto simp add: project_act_def)
374 lemma project_act_Restrict_Id_eq: "project_act h (Restrict C Id) = Restrict (project_set h C) Id"
375   by (auto simp add: project_act_def)
377 lemma project_extend_eq:
378      "project h C (extend h F) =
379       mk_program (Init F, Restrict (project_set h C) ` Acts F,
380                   {act. Restrict (project_set h C) act
381                           \<in> project_act h ` Restrict C `
382                                      (project_act h -` AllowedActs F)})"
383 apply (rule program_equalityI)
384   apply simp
385  apply (simp add: image_eq_UN)
386 apply (simp add: project_def)
387 done
389 lemma extend_inverse [simp]:
390      "project h UNIV (extend h F) = F"
391 apply (simp (no_asm_simp) add: project_extend_eq image_eq_UN
392           subset_UNIV [THEN subset_trans, THEN Restrict_triv])
393 apply (rule program_equalityI)
394 apply (simp_all (no_asm))
395 apply (subst insert_absorb)
396 apply (simp (no_asm) add: bexI [of _ Id])
397 apply auto
398 apply (rename_tac "act")
399 apply (rule_tac x = "extend_act h act" in bexI, auto)
400 done
402 lemma inj_extend: "inj (extend h)"
403 apply (rule inj_on_inverseI)
404 apply (rule extend_inverse)
405 done
407 lemma extend_Join [simp]: "extend h (F\<squnion>G) = extend h F\<squnion>extend h G"
408 apply (rule program_equalityI)
409 apply (simp (no_asm) add: extend_set_Int_distrib)
410 apply (simp add: image_Un, auto)
411 done
413 lemma extend_JN [simp]: "extend h (JOIN I F) = (\<Squnion>i \<in> I. extend h (F i))"
414 apply (rule program_equalityI)
415   apply (simp (no_asm) add: extend_set_INT_distrib)
416  apply (simp add: image_UN, auto)
417 done
419 (** These monotonicity results look natural but are UNUSED **)
421 lemma extend_mono: "F \<le> G ==> extend h F \<le> extend h G"
422   by (force simp add: component_eq_subset)
424 lemma project_mono: "F \<le> G ==> project h C F \<le> project h C G"
425   by (simp add: component_eq_subset, blast)
427 lemma all_total_extend: "all_total F ==> all_total (extend h F)"
428   by (simp add: all_total_def Domain_extend_act)
430 subsection{*Safety: co, stable*}
432 lemma extend_constrains:
433      "(extend h F \<in> (extend_set h A) co (extend_set h B)) =
434       (F \<in> A co B)"
435   by (simp add: constrains_def)
437 lemma extend_stable:
438      "(extend h F \<in> stable (extend_set h A)) = (F \<in> stable A)"
439   by (simp add: stable_def extend_constrains)
441 lemma extend_invariant:
442      "(extend h F \<in> invariant (extend_set h A)) = (F \<in> invariant A)"
443   by (simp add: invariant_def extend_stable)
445 (*Projects the state predicates in the property satisfied by  extend h F.
446   Converse fails: A and B may differ in their extra variables*)
447 lemma extend_constrains_project_set:
448      "extend h F \<in> A co B ==> F \<in> (project_set h A) co (project_set h B)"
449   by (auto simp add: constrains_def, force)
451 lemma extend_stable_project_set:
452      "extend h F \<in> stable A ==> F \<in> stable (project_set h A)"
453   by (simp add: stable_def extend_constrains_project_set)
456 subsection{*Weak safety primitives: Co, Stable*}
458 lemma reachable_extend_f: "p \<in> reachable (extend h F) ==> f p \<in> reachable F"
459   by (induct set: reachable) (auto intro: reachable.intros simp add: extend_act_def image_iff)
461 lemma h_reachable_extend: "h(s,y) \<in> reachable (extend h F) ==> s \<in> reachable F"
462   by (force dest!: reachable_extend_f)
464 lemma reachable_extend_eq: "reachable (extend h F) = extend_set h (reachable F)"
465 apply (unfold extend_set_def)
466 apply (rule equalityI)
467 apply (force intro: h_f_g_eq [symmetric] dest!: reachable_extend_f, clarify)
468 apply (erule reachable.induct)
469 apply (force intro: reachable.intros)+
470 done
472 lemma extend_Constrains:
473      "(extend h F \<in> (extend_set h A) Co (extend_set h B)) =
474       (F \<in> A Co B)"
475   by (simp add: Constrains_def reachable_extend_eq extend_constrains
476               extend_set_Int_distrib [symmetric])
478 lemma extend_Stable: "(extend h F \<in> Stable (extend_set h A)) = (F \<in> Stable A)"
479   by (simp add: Stable_def extend_Constrains)
481 lemma extend_Always: "(extend h F \<in> Always (extend_set h A)) = (F \<in> Always A)"
482   by (simp add: Always_def extend_Stable)
485 (** Safety and "project" **)
487 (** projection: monotonicity for safety **)
489 lemma (in -) project_act_mono:
490      "D \<subseteq> C ==>
491       project_act h (Restrict D act) \<subseteq> project_act h (Restrict C act)"
492   by (auto simp add: project_act_def)
494 lemma project_constrains_mono:
495      "[| D \<subseteq> C; project h C F \<in> A co B |] ==> project h D F \<in> A co B"
496 apply (auto simp add: constrains_def)
497 apply (drule project_act_mono, blast)
498 done
500 lemma project_stable_mono:
501      "[| D \<subseteq> C;  project h C F \<in> stable A |] ==> project h D F \<in> stable A"
502   by (simp add: stable_def project_constrains_mono)
504 (*Key lemma used in several proofs about project and co*)
505 lemma project_constrains:
506      "(project h C F \<in> A co B)  =
507       (F \<in> (C \<inter> extend_set h A) co (extend_set h B) & A \<subseteq> B)"
508 apply (unfold constrains_def)
509 apply (auto intro!: project_act_I simp add: ball_Un)
510 apply (force intro!: project_act_I dest!: subsetD)
511 (*the <== direction*)
512 apply (unfold project_act_def)
513 apply (force dest!: subsetD)
514 done
516 lemma project_stable: "(project h UNIV F \<in> stable A) = (F \<in> stable (extend_set h A))"
517   by (simp add: stable_def project_constrains)
519 lemma project_stable_I: "F \<in> stable (extend_set h A) ==> project h C F \<in> stable A"
520 apply (drule project_stable [THEN iffD2])
521 apply (blast intro: project_stable_mono)
522 done
524 lemma Int_extend_set_lemma:
525      "A \<inter> extend_set h ((project_set h A) \<inter> B) = A \<inter> extend_set h B"
526   by (auto simp add: split_extended_all)
528 (*Strange (look at occurrences of C) but used in leadsETo proofs*)
529 lemma project_constrains_project_set:
530      "G \<in> C co B ==> project h C G \<in> project_set h C co project_set h B"
531   by (simp add: constrains_def project_def project_act_def, blast)
533 lemma project_stable_project_set:
534      "G \<in> stable C ==> project h C G \<in> stable (project_set h C)"
535   by (simp add: stable_def project_constrains_project_set)
538 subsection{*Progress: transient, ensures*}
540 lemma extend_transient:
541      "(extend h F \<in> transient (extend_set h A)) = (F \<in> transient A)"
542   by (auto simp add: transient_def extend_set_subset_Compl_eq Domain_extend_act)
544 lemma extend_ensures:
545      "(extend h F \<in> (extend_set h A) ensures (extend_set h B)) =
546       (F \<in> A ensures B)"
547   by (simp add: ensures_def extend_constrains extend_transient
548         extend_set_Un_distrib [symmetric] extend_set_Diff_distrib [symmetric])
551      "F \<in> A leadsTo B
552       ==> extend h F \<in> (extend_set h A) leadsTo (extend_set h B)"
553 apply (erule leadsTo_induct)
555  apply (blast intro: leadsTo_Trans)
557 done
559 subsection{*Proving the converse takes some doing!*}
561 lemma slice_iff [iff]: "(x \<in> slice C y) = (h(x,y) \<in> C)"
562   by (simp add: slice_def)
564 lemma slice_Union: "slice (Union S) y = (\<Union>x \<in> S. slice x y)"
565   by auto
567 lemma slice_extend_set: "slice (extend_set h A) y = A"
568   by auto
570 lemma project_set_is_UN_slice: "project_set h A = (\<Union>y. slice A y)"
571   by auto
573 lemma extend_transient_slice:
574      "extend h F \<in> transient A ==> F \<in> transient (slice A y)"
575   by (auto simp: transient_def)
577 (*Converse?*)
578 lemma extend_constrains_slice:
579      "extend h F \<in> A co B ==> F \<in> (slice A y) co (slice B y)"
580   by (auto simp add: constrains_def)
582 lemma extend_ensures_slice:
583      "extend h F \<in> A ensures B ==> F \<in> (slice A y) ensures (project_set h B)"
584 apply (auto simp add: ensures_def extend_constrains extend_transient)
585 apply (erule_tac  extend_transient_slice [THEN transient_strengthen])
586 apply (erule extend_constrains_slice [THEN constrains_weaken], auto)
587 done
590      "\<forall>y. F \<in> (slice B y) leadsTo CU ==> F \<in> (project_set h B) leadsTo CU"
591 apply (simp add: project_set_is_UN_slice)
592 apply (blast intro: leadsTo_UN)
593 done
595 lemma extend_leadsTo_slice [rule_format]:
596      "extend h F \<in> AU leadsTo BU
597       ==> \<forall>y. F \<in> (slice AU y) leadsTo (project_set h BU)"
598 apply (erule leadsTo_induct)
599   apply (blast intro: extend_ensures_slice)
602 done
605      "(extend h F \<in> (extend_set h A) leadsTo (extend_set h B)) =
606       (F \<in> A leadsTo B)"
607 apply safe
609 apply (drule extend_leadsTo_slice)
610 apply (simp add: slice_extend_set)
611 done
614      "(extend h F \<in> (extend_set h A) LeadsTo (extend_set h B)) =
615       (F \<in> A LeadsTo B)"
617               extend_set_Int_distrib [symmetric])
620 subsection{*preserves*}
622 lemma project_preserves_I:
623      "G \<in> preserves (v o f) ==> project h C G \<in> preserves v"
624   by (auto simp add: preserves_def project_stable_I extend_set_eq_Collect)
626 (*to preserve f is to preserve the whole original state*)
627 lemma project_preserves_id_I:
628      "G \<in> preserves f ==> project h C G \<in> preserves id"
629   by (simp add: project_preserves_I)
631 lemma extend_preserves:
632      "(extend h G \<in> preserves (v o f)) = (G \<in> preserves v)"
633   by (auto simp add: preserves_def extend_stable [symmetric]
634                    extend_set_eq_Collect)
636 lemma inj_extend_preserves: "inj h ==> (extend h G \<in> preserves g)"
637   by (auto simp add: preserves_def extend_def extend_act_def stable_def
638                    constrains_def g_def)
641 subsection{*Guarantees*}
643 lemma project_extend_Join: "project h UNIV ((extend h F)\<squnion>G) = F\<squnion>(project h UNIV G)"
644 apply (rule program_equalityI)
645   apply (simp add: project_set_extend_set_Int)
646  apply (auto simp add: image_eq_UN)
647 done
649 lemma extend_Join_eq_extend_D:
650      "(extend h F)\<squnion>G = extend h H ==> H = F\<squnion>(project h UNIV G)"
651 apply (drule_tac f = "project h UNIV" in arg_cong)
652 apply (simp add: project_extend_Join)
653 done
655 (** Strong precondition and postcondition; only useful when
656     the old and new state sets are in bijection **)
659 lemma ok_extend_imp_ok_project: "extend h F ok G ==> F ok project h UNIV G"
660 apply (auto simp add: ok_def)
661 apply (drule subsetD)
662 apply (auto intro!: rev_image_eqI)
663 done
665 lemma ok_extend_iff: "(extend h F ok extend h G) = (F ok G)"
666 apply (simp add: ok_def, safe)
667 apply force+
668 done
670 lemma OK_extend_iff: "OK I (%i. extend h (F i)) = (OK I F)"
671 apply (unfold OK_def, safe)
672 apply (drule_tac x = i in bspec)
673 apply (drule_tac  x = j in bspec)
674 apply force+
675 done
677 lemma guarantees_imp_extend_guarantees:
678      "F \<in> X guarantees Y ==>
679       extend h F \<in> (extend h ` X) guarantees (extend h ` Y)"
680 apply (rule guaranteesI, clarify)
681 apply (blast dest: ok_extend_imp_ok_project extend_Join_eq_extend_D
682                    guaranteesD)
683 done
685 lemma extend_guarantees_imp_guarantees:
686      "extend h F \<in> (extend h ` X) guarantees (extend h ` Y)
687       ==> F \<in> X guarantees Y"
688 apply (auto simp add: guar_def)
689 apply (drule_tac x = "extend h G" in spec)
690 apply (simp del: extend_Join
691             add: extend_Join [symmetric] ok_extend_iff
692                  inj_extend [THEN inj_image_mem_iff])
693 done
695 lemma extend_guarantees_eq:
696      "(extend h F \<in> (extend h ` X) guarantees (extend h ` Y)) =
697       (F \<in> X guarantees Y)"
698   by (blast intro: guarantees_imp_extend_guarantees
699                  extend_guarantees_imp_guarantees)
701 end
703 end