src/HOL/UNITY/Guar.thy
author wenzelm
Mon Mar 16 18:24:30 2009 +0100 (2009-03-16)
changeset 30549 d2d7874648bd
parent 27682 25aceefd4786
child 32960 69916a850301
permissions -rw-r--r--
simplified method setup;
     1 (*  Title:      HOL/UNITY/Guar.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1999  University of Cambridge
     5 
     6 From Chandy and Sanders, "Reasoning About Program Composition",
     7 Technical Report 2000-003, University of Florida, 2000.
     8 
     9 Revised by Sidi Ehmety on January 2001
    10 
    11 Added: Compatibility, weakest guarantees, etc.
    12 
    13 and Weakest existential property,
    14 from Charpentier and Chandy "Theorems about Composition",
    15 Fifth International Conference on Mathematics of Program, 2000.
    16 
    17 *)
    18 
    19 header{*Guarantees Specifications*}
    20 
    21 theory Guar
    22 imports Comp
    23 begin
    24 
    25 instance program :: (type) order
    26 proof qed (auto simp add: program_less_le dest: component_antisym intro: component_refl component_trans)
    27 
    28 text{*Existential and Universal properties.  I formalize the two-program
    29       case, proving equivalence with Chandy and Sanders's n-ary definitions*}
    30 
    31 constdefs
    32 
    33   ex_prop  :: "'a program set => bool"
    34    "ex_prop X == \<forall>F G. F ok G -->F \<in> X | G \<in> X --> (F\<squnion>G) \<in> X"
    35 
    36   strict_ex_prop  :: "'a program set => bool"
    37    "strict_ex_prop X == \<forall>F G.  F ok G --> (F \<in> X | G \<in> X) = (F\<squnion>G \<in> X)"
    38 
    39   uv_prop  :: "'a program set => bool"
    40    "uv_prop X == SKIP \<in> X & (\<forall>F G. F ok G --> F \<in> X & G \<in> X --> (F\<squnion>G) \<in> X)"
    41 
    42   strict_uv_prop  :: "'a program set => bool"
    43    "strict_uv_prop X == 
    44       SKIP \<in> X & (\<forall>F G. F ok G --> (F \<in> X & G \<in> X) = (F\<squnion>G \<in> X))"
    45 
    46 
    47 text{*Guarantees properties*}
    48 
    49 constdefs
    50 
    51   guar :: "['a program set, 'a program set] => 'a program set"
    52           (infixl "guarantees" 55)  (*higher than membership, lower than Co*)
    53    "X guarantees Y == {F. \<forall>G. F ok G --> F\<squnion>G \<in> X --> F\<squnion>G \<in> Y}"
    54   
    55 
    56   (* Weakest guarantees *)
    57    wg :: "['a program, 'a program set] =>  'a program set"
    58   "wg F Y == Union({X. F \<in> X guarantees Y})"
    59 
    60    (* Weakest existential property stronger than X *)
    61    wx :: "('a program) set => ('a program)set"
    62    "wx X == Union({Y. Y \<subseteq> X & ex_prop Y})"
    63   
    64   (*Ill-defined programs can arise through "Join"*)
    65   welldef :: "'a program set"
    66   "welldef == {F. Init F \<noteq> {}}"
    67   
    68   refines :: "['a program, 'a program, 'a program set] => bool"
    69 			("(3_ refines _ wrt _)" [10,10,10] 10)
    70   "G refines F wrt X ==
    71      \<forall>H. (F ok H & G ok H & F\<squnion>H \<in> welldef \<inter> X) --> 
    72          (G\<squnion>H \<in> welldef \<inter> X)"
    73 
    74   iso_refines :: "['a program, 'a program, 'a program set] => bool"
    75                               ("(3_ iso'_refines _ wrt _)" [10,10,10] 10)
    76   "G iso_refines F wrt X ==
    77    F \<in> welldef \<inter> X --> G \<in> welldef \<inter> X"
    78 
    79 
    80 lemma OK_insert_iff:
    81      "(OK (insert i I) F) = 
    82       (if i \<in> I then OK I F else OK I F & (F i ok JOIN I F))"
    83 by (auto intro: ok_sym simp add: OK_iff_ok)
    84 
    85 
    86 subsection{*Existential Properties*}
    87 
    88 lemma ex1 [rule_format]: 
    89  "[| ex_prop X; finite GG |] ==>  
    90      GG \<inter> X \<noteq> {}--> OK GG (%G. G) --> (\<Squnion>G \<in> GG. G) \<in> X"
    91 apply (unfold ex_prop_def)
    92 apply (erule finite_induct)
    93 apply (auto simp add: OK_insert_iff Int_insert_left)
    94 done
    95 
    96 
    97 lemma ex2: 
    98      "\<forall>GG. finite GG & GG \<inter> X \<noteq> {} --> OK GG (%G. G) -->(\<Squnion>G \<in> GG. G):X 
    99       ==> ex_prop X"
   100 apply (unfold ex_prop_def, clarify)
   101 apply (drule_tac x = "{F,G}" in spec)
   102 apply (auto dest: ok_sym simp add: OK_iff_ok)
   103 done
   104 
   105 
   106 (*Chandy & Sanders take this as a definition*)
   107 lemma ex_prop_finite:
   108      "ex_prop X = 
   109       (\<forall>GG. finite GG & GG \<inter> X \<noteq> {} & OK GG (%G. G)--> (\<Squnion>G \<in> GG. G) \<in> X)"
   110 by (blast intro: ex1 ex2)
   111 
   112 
   113 (*Their "equivalent definition" given at the end of section 3*)
   114 lemma ex_prop_equiv: 
   115      "ex_prop X = (\<forall>G. G \<in> X = (\<forall>H. (G component_of H) --> H \<in> X))"
   116 apply auto
   117 apply (unfold ex_prop_def component_of_def, safe, blast, blast) 
   118 apply (subst Join_commute) 
   119 apply (drule ok_sym, blast) 
   120 done
   121 
   122 
   123 subsection{*Universal Properties*}
   124 
   125 lemma uv1 [rule_format]: 
   126      "[| uv_prop X; finite GG |] 
   127       ==> GG \<subseteq> X & OK GG (%G. G) --> (\<Squnion>G \<in> GG. G) \<in> X"
   128 apply (unfold uv_prop_def)
   129 apply (erule finite_induct)
   130 apply (auto simp add: Int_insert_left OK_insert_iff)
   131 done
   132 
   133 lemma uv2: 
   134      "\<forall>GG. finite GG & GG \<subseteq> X & OK GG (%G. G) --> (\<Squnion>G \<in> GG. G) \<in> X  
   135       ==> uv_prop X"
   136 apply (unfold uv_prop_def)
   137 apply (rule conjI)
   138  apply (drule_tac x = "{}" in spec)
   139  prefer 2
   140  apply clarify 
   141  apply (drule_tac x = "{F,G}" in spec)
   142 apply (auto dest: ok_sym simp add: OK_iff_ok)
   143 done
   144 
   145 (*Chandy & Sanders take this as a definition*)
   146 lemma uv_prop_finite:
   147      "uv_prop X = 
   148       (\<forall>GG. finite GG & GG \<subseteq> X & OK GG (%G. G) --> (\<Squnion>G \<in> GG. G): X)"
   149 by (blast intro: uv1 uv2)
   150 
   151 subsection{*Guarantees*}
   152 
   153 lemma guaranteesI:
   154      "(!!G. [| F ok G; F\<squnion>G \<in> X |] ==> F\<squnion>G \<in> Y) ==> F \<in> X guarantees Y"
   155 by (simp add: guar_def component_def)
   156 
   157 lemma guaranteesD: 
   158      "[| F \<in> X guarantees Y;  F ok G;  F\<squnion>G \<in> X |] ==> F\<squnion>G \<in> Y"
   159 by (unfold guar_def component_def, blast)
   160 
   161 (*This version of guaranteesD matches more easily in the conclusion
   162   The major premise can no longer be  F \<subseteq> H since we need to reason about G*)
   163 lemma component_guaranteesD: 
   164      "[| F \<in> X guarantees Y;  F\<squnion>G = H;  H \<in> X;  F ok G |] ==> H \<in> Y"
   165 by (unfold guar_def, blast)
   166 
   167 lemma guarantees_weaken: 
   168      "[| F \<in> X guarantees X'; Y \<subseteq> X; X' \<subseteq> Y' |] ==> F \<in> Y guarantees Y'"
   169 by (unfold guar_def, blast)
   170 
   171 lemma subset_imp_guarantees_UNIV: "X \<subseteq> Y ==> X guarantees Y = UNIV"
   172 by (unfold guar_def, blast)
   173 
   174 (*Equivalent to subset_imp_guarantees_UNIV but more intuitive*)
   175 lemma subset_imp_guarantees: "X \<subseteq> Y ==> F \<in> X guarantees Y"
   176 by (unfold guar_def, blast)
   177 
   178 (*Remark at end of section 4.1 *)
   179 
   180 lemma ex_prop_imp: "ex_prop Y ==> (Y = UNIV guarantees Y)"
   181 apply (simp (no_asm_use) add: guar_def ex_prop_equiv)
   182 apply safe
   183  apply (drule_tac x = x in spec)
   184  apply (drule_tac [2] x = x in spec)
   185  apply (drule_tac [2] sym)
   186 apply (auto simp add: component_of_def)
   187 done
   188 
   189 lemma guarantees_imp: "(Y = UNIV guarantees Y) ==> ex_prop(Y)"
   190 by (auto simp add: guar_def ex_prop_equiv component_of_def dest: sym)
   191 
   192 lemma ex_prop_equiv2: "(ex_prop Y) = (Y = UNIV guarantees Y)"
   193 apply (rule iffI)
   194 apply (rule ex_prop_imp)
   195 apply (auto simp add: guarantees_imp) 
   196 done
   197 
   198 
   199 subsection{*Distributive Laws.  Re-Orient to Perform Miniscoping*}
   200 
   201 lemma guarantees_UN_left: 
   202      "(\<Union>i \<in> I. X i) guarantees Y = (\<Inter>i \<in> I. X i guarantees Y)"
   203 by (unfold guar_def, blast)
   204 
   205 lemma guarantees_Un_left: 
   206      "(X \<union> Y) guarantees Z = (X guarantees Z) \<inter> (Y guarantees Z)"
   207 by (unfold guar_def, blast)
   208 
   209 lemma guarantees_INT_right: 
   210      "X guarantees (\<Inter>i \<in> I. Y i) = (\<Inter>i \<in> I. X guarantees Y i)"
   211 by (unfold guar_def, blast)
   212 
   213 lemma guarantees_Int_right: 
   214      "Z guarantees (X \<inter> Y) = (Z guarantees X) \<inter> (Z guarantees Y)"
   215 by (unfold guar_def, blast)
   216 
   217 lemma guarantees_Int_right_I:
   218      "[| F \<in> Z guarantees X;  F \<in> Z guarantees Y |]  
   219      ==> F \<in> Z guarantees (X \<inter> Y)"
   220 by (simp add: guarantees_Int_right)
   221 
   222 lemma guarantees_INT_right_iff:
   223      "(F \<in> X guarantees (INTER I Y)) = (\<forall>i\<in>I. F \<in> X guarantees (Y i))"
   224 by (simp add: guarantees_INT_right)
   225 
   226 lemma shunting: "(X guarantees Y) = (UNIV guarantees (-X \<union> Y))"
   227 by (unfold guar_def, blast)
   228 
   229 lemma contrapositive: "(X guarantees Y) = -Y guarantees -X"
   230 by (unfold guar_def, blast)
   231 
   232 (** The following two can be expressed using intersection and subset, which
   233     is more faithful to the text but looks cryptic.
   234 **)
   235 
   236 lemma combining1: 
   237     "[| F \<in> V guarantees X;  F \<in> (X \<inter> Y) guarantees Z |] 
   238      ==> F \<in> (V \<inter> Y) guarantees Z"
   239 by (unfold guar_def, blast)
   240 
   241 lemma combining2: 
   242     "[| F \<in> V guarantees (X \<union> Y);  F \<in> Y guarantees Z |] 
   243      ==> F \<in> V guarantees (X \<union> Z)"
   244 by (unfold guar_def, blast)
   245 
   246 (** The following two follow Chandy-Sanders, but the use of object-quantifiers
   247     does not suit Isabelle... **)
   248 
   249 (*Premise should be (!!i. i \<in> I ==> F \<in> X guarantees Y i) *)
   250 lemma all_guarantees: 
   251      "\<forall>i\<in>I. F \<in> X guarantees (Y i) ==> F \<in> X guarantees (\<Inter>i \<in> I. Y i)"
   252 by (unfold guar_def, blast)
   253 
   254 (*Premises should be [| F \<in> X guarantees Y i; i \<in> I |] *)
   255 lemma ex_guarantees: 
   256      "\<exists>i\<in>I. F \<in> X guarantees (Y i) ==> F \<in> X guarantees (\<Union>i \<in> I. Y i)"
   257 by (unfold guar_def, blast)
   258 
   259 
   260 subsection{*Guarantees: Additional Laws (by lcp)*}
   261 
   262 lemma guarantees_Join_Int: 
   263     "[| F \<in> U guarantees V;  G \<in> X guarantees Y; F ok G |]  
   264      ==> F\<squnion>G \<in> (U \<inter> X) guarantees (V \<inter> Y)"
   265 apply (simp add: guar_def, safe)
   266  apply (simp add: Join_assoc)
   267 apply (subgoal_tac "F\<squnion>G\<squnion>Ga = G\<squnion>(F\<squnion>Ga) ")
   268  apply (simp add: ok_commute)
   269 apply (simp add: Join_ac)
   270 done
   271 
   272 lemma guarantees_Join_Un: 
   273     "[| F \<in> U guarantees V;  G \<in> X guarantees Y; F ok G |]   
   274      ==> F\<squnion>G \<in> (U \<union> X) guarantees (V \<union> Y)"
   275 apply (simp add: guar_def, safe)
   276  apply (simp add: Join_assoc)
   277 apply (subgoal_tac "F\<squnion>G\<squnion>Ga = G\<squnion>(F\<squnion>Ga) ")
   278  apply (simp add: ok_commute)
   279 apply (simp add: Join_ac)
   280 done
   281 
   282 lemma guarantees_JN_INT: 
   283      "[| \<forall>i\<in>I. F i \<in> X i guarantees Y i;  OK I F |]  
   284       ==> (JOIN I F) \<in> (INTER I X) guarantees (INTER I Y)"
   285 apply (unfold guar_def, auto)
   286 apply (drule bspec, assumption)
   287 apply (rename_tac "i")
   288 apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
   289 apply (auto intro: OK_imp_ok
   290             simp add: Join_assoc [symmetric] JN_Join_diff JN_absorb)
   291 done
   292 
   293 lemma guarantees_JN_UN: 
   294     "[| \<forall>i\<in>I. F i \<in> X i guarantees Y i;  OK I F |]  
   295      ==> (JOIN I F) \<in> (UNION I X) guarantees (UNION I Y)"
   296 apply (unfold guar_def, auto)
   297 apply (drule bspec, assumption)
   298 apply (rename_tac "i")
   299 apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
   300 apply (auto intro: OK_imp_ok
   301             simp add: Join_assoc [symmetric] JN_Join_diff JN_absorb)
   302 done
   303 
   304 
   305 subsection{*Guarantees Laws for Breaking Down the Program (by lcp)*}
   306 
   307 lemma guarantees_Join_I1: 
   308      "[| F \<in> X guarantees Y;  F ok G |] ==> F\<squnion>G \<in> X guarantees Y"
   309 by (simp add: guar_def Join_assoc)
   310 
   311 lemma guarantees_Join_I2:         
   312      "[| G \<in> X guarantees Y;  F ok G |] ==> F\<squnion>G \<in> X guarantees Y"
   313 apply (simp add: Join_commute [of _ G] ok_commute [of _ G])
   314 apply (blast intro: guarantees_Join_I1)
   315 done
   316 
   317 lemma guarantees_JN_I: 
   318      "[| i \<in> I;  F i \<in> X guarantees Y;  OK I F |]  
   319       ==> (\<Squnion>i \<in> I. (F i)) \<in> X guarantees Y"
   320 apply (unfold guar_def, clarify)
   321 apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
   322 apply (auto intro: OK_imp_ok 
   323             simp add: JN_Join_diff JN_Join_diff Join_assoc [symmetric])
   324 done
   325 
   326 
   327 (*** well-definedness ***)
   328 
   329 lemma Join_welldef_D1: "F\<squnion>G \<in> welldef ==> F \<in> welldef"
   330 by (unfold welldef_def, auto)
   331 
   332 lemma Join_welldef_D2: "F\<squnion>G \<in> welldef ==> G \<in> welldef"
   333 by (unfold welldef_def, auto)
   334 
   335 (*** refinement ***)
   336 
   337 lemma refines_refl: "F refines F wrt X"
   338 by (unfold refines_def, blast)
   339 
   340 (*We'd like transitivity, but how do we get it?*)
   341 lemma refines_trans:
   342      "[| H refines G wrt X;  G refines F wrt X |] ==> H refines F wrt X"
   343 apply (simp add: refines_def) 
   344 oops
   345 
   346 
   347 lemma strict_ex_refine_lemma: 
   348      "strict_ex_prop X  
   349       ==> (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> X --> G\<squnion>H \<in> X)  
   350               = (F \<in> X --> G \<in> X)"
   351 by (unfold strict_ex_prop_def, auto)
   352 
   353 lemma strict_ex_refine_lemma_v: 
   354      "strict_ex_prop X  
   355       ==> (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef & F\<squnion>H \<in> X --> G\<squnion>H \<in> X) =  
   356           (F \<in> welldef \<inter> X --> G \<in> X)"
   357 apply (unfold strict_ex_prop_def, safe)
   358 apply (erule_tac x = SKIP and P = "%H. ?PP H --> ?RR H" in allE)
   359 apply (auto dest: Join_welldef_D1 Join_welldef_D2)
   360 done
   361 
   362 lemma ex_refinement_thm:
   363      "[| strict_ex_prop X;   
   364          \<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef \<inter> X --> G\<squnion>H \<in> welldef |]  
   365       ==> (G refines F wrt X) = (G iso_refines F wrt X)"
   366 apply (rule_tac x = SKIP in allE, assumption)
   367 apply (simp add: refines_def iso_refines_def strict_ex_refine_lemma_v)
   368 done
   369 
   370 
   371 lemma strict_uv_refine_lemma: 
   372      "strict_uv_prop X ==> 
   373       (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> X --> G\<squnion>H \<in> X) = (F \<in> X --> G \<in> X)"
   374 by (unfold strict_uv_prop_def, blast)
   375 
   376 lemma strict_uv_refine_lemma_v: 
   377      "strict_uv_prop X  
   378       ==> (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef & F\<squnion>H \<in> X --> G\<squnion>H \<in> X) =  
   379           (F \<in> welldef \<inter> X --> G \<in> X)"
   380 apply (unfold strict_uv_prop_def, safe)
   381 apply (erule_tac x = SKIP and P = "%H. ?PP H --> ?RR H" in allE)
   382 apply (auto dest: Join_welldef_D1 Join_welldef_D2)
   383 done
   384 
   385 lemma uv_refinement_thm:
   386      "[| strict_uv_prop X;   
   387          \<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef \<inter> X --> 
   388              G\<squnion>H \<in> welldef |]  
   389       ==> (G refines F wrt X) = (G iso_refines F wrt X)"
   390 apply (rule_tac x = SKIP in allE, assumption)
   391 apply (simp add: refines_def iso_refines_def strict_uv_refine_lemma_v)
   392 done
   393 
   394 (* Added by Sidi Ehmety from Chandy & Sander, section 6 *)
   395 lemma guarantees_equiv: 
   396     "(F \<in> X guarantees Y) = (\<forall>H. H \<in> X \<longrightarrow> (F component_of H \<longrightarrow> H \<in> Y))"
   397 by (unfold guar_def component_of_def, auto)
   398 
   399 lemma wg_weakest: "!!X. F\<in> (X guarantees Y) ==> X \<subseteq> (wg F Y)"
   400 by (unfold wg_def, auto)
   401 
   402 lemma wg_guarantees: "F\<in> ((wg F Y) guarantees Y)"
   403 by (unfold wg_def guar_def, blast)
   404 
   405 lemma wg_equiv: "(H \<in> wg F X) = (F component_of H --> H \<in> X)"
   406 by (simp add: guarantees_equiv wg_def, blast)
   407 
   408 lemma component_of_wg: "F component_of H ==> (H \<in> wg F X) = (H \<in> X)"
   409 by (simp add: wg_equiv)
   410 
   411 lemma wg_finite: 
   412     "\<forall>FF. finite FF & FF \<inter> X \<noteq> {} --> OK FF (%F. F)  
   413           --> (\<forall>F\<in>FF. ((\<Squnion>F \<in> FF. F): wg F X) = ((\<Squnion>F \<in> FF. F):X))"
   414 apply clarify
   415 apply (subgoal_tac "F component_of (\<Squnion>F \<in> FF. F) ")
   416 apply (drule_tac X = X in component_of_wg, simp)
   417 apply (simp add: component_of_def)
   418 apply (rule_tac x = "\<Squnion>F \<in> (FF-{F}) . F" in exI)
   419 apply (auto intro: JN_Join_diff dest: ok_sym simp add: OK_iff_ok)
   420 done
   421 
   422 lemma wg_ex_prop: "ex_prop X ==> (F \<in> X) = (\<forall>H. H \<in> wg F X)"
   423 apply (simp (no_asm_use) add: ex_prop_equiv wg_equiv)
   424 apply blast
   425 done
   426 
   427 (** From Charpentier and Chandy "Theorems About Composition" **)
   428 (* Proposition 2 *)
   429 lemma wx_subset: "(wx X)<=X"
   430 by (unfold wx_def, auto)
   431 
   432 lemma wx_ex_prop: "ex_prop (wx X)"
   433 apply (simp add: wx_def ex_prop_equiv cong: bex_cong, safe, blast)
   434 apply force 
   435 done
   436 
   437 lemma wx_weakest: "\<forall>Z. Z<= X --> ex_prop Z --> Z \<subseteq> wx X"
   438 by (auto simp add: wx_def)
   439 
   440 (* Proposition 6 *)
   441 lemma wx'_ex_prop: "ex_prop({F. \<forall>G. F ok G --> F\<squnion>G \<in> X})"
   442 apply (unfold ex_prop_def, safe)
   443  apply (drule_tac x = "G\<squnion>Ga" in spec)
   444  apply (force simp add: ok_Join_iff1 Join_assoc)
   445 apply (drule_tac x = "F\<squnion>Ga" in spec)
   446 apply (simp add: ok_Join_iff1 ok_commute  Join_ac) 
   447 done
   448 
   449 text{* Equivalence with the other definition of wx *}
   450 
   451 lemma wx_equiv: "wx X = {F. \<forall>G. F ok G --> (F\<squnion>G) \<in> X}"
   452 apply (unfold wx_def, safe)
   453  apply (simp add: ex_prop_def, blast) 
   454 apply (simp (no_asm))
   455 apply (rule_tac x = "{F. \<forall>G. F ok G --> F\<squnion>G \<in> X}" in exI, safe)
   456 apply (rule_tac [2] wx'_ex_prop)
   457 apply (drule_tac x = SKIP in spec)+
   458 apply auto 
   459 done
   460 
   461 
   462 text{* Propositions 7 to 11 are about this second definition of wx. 
   463    They are the same as the ones proved for the first definition of wx,
   464  by equivalence *}
   465    
   466 (* Proposition 12 *)
   467 (* Main result of the paper *)
   468 lemma guarantees_wx_eq: "(X guarantees Y) = wx(-X \<union> Y)"
   469 by (simp add: guar_def wx_equiv)
   470 
   471 
   472 (* Rules given in section 7 of Chandy and Sander's
   473     Reasoning About Program composition paper *)
   474 lemma stable_guarantees_Always:
   475      "Init F \<subseteq> A ==> F \<in> (stable A) guarantees (Always A)"
   476 apply (rule guaranteesI)
   477 apply (simp add: Join_commute)
   478 apply (rule stable_Join_Always1)
   479  apply (simp_all add: invariant_def Join_stable)
   480 done
   481 
   482 lemma constrains_guarantees_leadsTo:
   483      "F \<in> transient A ==> F \<in> (A co A \<union> B) guarantees (A leadsTo (B-A))"
   484 apply (rule guaranteesI)
   485 apply (rule leadsTo_Basis')
   486  apply (drule constrains_weaken_R)
   487   prefer 2 apply assumption
   488  apply blast
   489 apply (blast intro: Join_transient_I1)
   490 done
   491 
   492 end