src/HOL/UNITY/Constrains.thy
author wenzelm
Thu Jul 23 22:13:42 2015 +0200 (2015-07-23)
changeset 60773 d09c66a0ea10
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     1 (*  Title:      HOL/UNITY/Constrains.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1998  University of Cambridge
     4 
     5 Weak safety relations: restricted to the set of reachable states.
     6 *)
     7 
     8 section{*Weak Safety*}
     9 
    10 theory Constrains imports UNITY begin
    11 
    12   (*Initial states and program => (final state, reversed trace to it)...
    13     Arguments MUST be curried in an inductive definition*)
    14 
    15 inductive_set
    16   traces :: "['a set, ('a * 'a)set set] => ('a * 'a list) set"
    17   for init :: "'a set" and acts :: "('a * 'a)set set"
    18   where
    19          (*Initial trace is empty*)
    20     Init:  "s \<in> init ==> (s,[]) \<in> traces init acts"
    21 
    22   | Acts:  "[| act: acts;  (s,evs) \<in> traces init acts;  (s,s'): act |]
    23             ==> (s', s#evs) \<in> traces init acts"
    24 
    25 
    26 inductive_set
    27   reachable :: "'a program => 'a set"
    28   for F :: "'a program"
    29   where
    30     Init:  "s \<in> Init F ==> s \<in> reachable F"
    31 
    32   | Acts:  "[| act: Acts F;  s \<in> reachable F;  (s,s'): act |]
    33             ==> s' \<in> reachable F"
    34 
    35 definition Constrains :: "['a set, 'a set] => 'a program set" (infixl "Co" 60) where
    36     "A Co B == {F. F \<in> (reachable F \<inter> A)  co  B}"
    37 
    38 definition Unless  :: "['a set, 'a set] => 'a program set" (infixl "Unless" 60) where
    39     "A Unless B == (A-B) Co (A \<union> B)"
    40 
    41 definition Stable     :: "'a set => 'a program set" where
    42     "Stable A == A Co A"
    43 
    44   (*Always is the weak form of "invariant"*)
    45 definition Always :: "'a set => 'a program set" where
    46     "Always A == {F. Init F \<subseteq> A} \<inter> Stable A"
    47 
    48   (*Polymorphic in both states and the meaning of \<le> *)
    49 definition Increasing :: "['a => 'b::{order}] => 'a program set" where
    50     "Increasing f == \<Inter>z. Stable {s. z \<le> f s}"
    51 
    52 
    53 subsection{*traces and reachable*}
    54 
    55 lemma reachable_equiv_traces:
    56      "reachable F = {s. \<exists>evs. (s,evs) \<in> traces (Init F) (Acts F)}"
    57 apply safe
    58 apply (erule_tac [2] traces.induct)
    59 apply (erule reachable.induct)
    60 apply (blast intro: reachable.intros traces.intros)+
    61 done
    62 
    63 lemma Init_subset_reachable: "Init F \<subseteq> reachable F"
    64 by (blast intro: reachable.intros)
    65 
    66 lemma stable_reachable [intro!,simp]:
    67      "Acts G \<subseteq> Acts F ==> G \<in> stable (reachable F)"
    68 by (blast intro: stableI constrainsI reachable.intros)
    69 
    70 (*The set of all reachable states is an invariant...*)
    71 lemma invariant_reachable: "F \<in> invariant (reachable F)"
    72 apply (simp add: invariant_def)
    73 apply (blast intro: reachable.intros)
    74 done
    75 
    76 (*...in fact the strongest invariant!*)
    77 lemma invariant_includes_reachable: "F \<in> invariant A ==> reachable F \<subseteq> A"
    78 apply (simp add: stable_def constrains_def invariant_def)
    79 apply (rule subsetI)
    80 apply (erule reachable.induct)
    81 apply (blast intro: reachable.intros)+
    82 done
    83 
    84 
    85 subsection{*Co*}
    86 
    87 (*F \<in> B co B' ==> F \<in> (reachable F \<inter> B) co (reachable F \<inter> B')*)
    88 lemmas constrains_reachable_Int =  
    89     subset_refl [THEN stable_reachable [unfolded stable_def], THEN constrains_Int]
    90 
    91 (*Resembles the previous definition of Constrains*)
    92 lemma Constrains_eq_constrains: 
    93      "A Co B = {F. F \<in> (reachable F  \<inter>  A) co (reachable F  \<inter>  B)}"
    94 apply (unfold Constrains_def)
    95 apply (blast dest: constrains_reachable_Int intro: constrains_weaken)
    96 done
    97 
    98 lemma constrains_imp_Constrains: "F \<in> A co A' ==> F \<in> A Co A'"
    99 apply (unfold Constrains_def)
   100 apply (blast intro: constrains_weaken_L)
   101 done
   102 
   103 lemma stable_imp_Stable: "F \<in> stable A ==> F \<in> Stable A"
   104 apply (unfold stable_def Stable_def)
   105 apply (erule constrains_imp_Constrains)
   106 done
   107 
   108 lemma ConstrainsI: 
   109     "(!!act s s'. [| act: Acts F;  (s,s') \<in> act;  s \<in> A |] ==> s': A')  
   110      ==> F \<in> A Co A'"
   111 apply (rule constrains_imp_Constrains)
   112 apply (blast intro: constrainsI)
   113 done
   114 
   115 lemma Constrains_empty [iff]: "F \<in> {} Co B"
   116 by (unfold Constrains_def constrains_def, blast)
   117 
   118 lemma Constrains_UNIV [iff]: "F \<in> A Co UNIV"
   119 by (blast intro: ConstrainsI)
   120 
   121 lemma Constrains_weaken_R: 
   122     "[| F \<in> A Co A'; A'<=B' |] ==> F \<in> A Co B'"
   123 apply (unfold Constrains_def)
   124 apply (blast intro: constrains_weaken_R)
   125 done
   126 
   127 lemma Constrains_weaken_L: 
   128     "[| F \<in> A Co A'; B \<subseteq> A |] ==> F \<in> B Co A'"
   129 apply (unfold Constrains_def)
   130 apply (blast intro: constrains_weaken_L)
   131 done
   132 
   133 lemma Constrains_weaken: 
   134    "[| F \<in> A Co A'; B \<subseteq> A; A'<=B' |] ==> F \<in> B Co B'"
   135 apply (unfold Constrains_def)
   136 apply (blast intro: constrains_weaken)
   137 done
   138 
   139 (** Union **)
   140 
   141 lemma Constrains_Un: 
   142     "[| F \<in> A Co A'; F \<in> B Co B' |] ==> F \<in> (A \<union> B) Co (A' \<union> B')"
   143 apply (unfold Constrains_def)
   144 apply (blast intro: constrains_Un [THEN constrains_weaken])
   145 done
   146 
   147 lemma Constrains_UN: 
   148   assumes Co: "!!i. i \<in> I ==> F \<in> (A i) Co (A' i)"
   149   shows "F \<in> (\<Union>i \<in> I. A i) Co (\<Union>i \<in> I. A' i)"
   150 apply (unfold Constrains_def)
   151 apply (rule CollectI)
   152 apply (rule Co [unfolded Constrains_def, THEN CollectD, THEN constrains_UN, 
   153                 THEN constrains_weaken],   auto)
   154 done
   155 
   156 (** Intersection **)
   157 
   158 lemma Constrains_Int: 
   159     "[| F \<in> A Co A'; F \<in> B Co B' |] ==> F \<in> (A \<inter> B) Co (A' \<inter> B')"
   160 apply (unfold Constrains_def)
   161 apply (blast intro: constrains_Int [THEN constrains_weaken])
   162 done
   163 
   164 lemma Constrains_INT: 
   165   assumes Co: "!!i. i \<in> I ==> F \<in> (A i) Co (A' i)"
   166   shows "F \<in> (\<Inter>i \<in> I. A i) Co (\<Inter>i \<in> I. A' i)"
   167 apply (unfold Constrains_def)
   168 apply (rule CollectI)
   169 apply (rule Co [unfolded Constrains_def, THEN CollectD, THEN constrains_INT, 
   170                 THEN constrains_weaken],   auto)
   171 done
   172 
   173 lemma Constrains_imp_subset: "F \<in> A Co A' ==> reachable F \<inter> A \<subseteq> A'"
   174 by (simp add: constrains_imp_subset Constrains_def)
   175 
   176 lemma Constrains_trans: "[| F \<in> A Co B; F \<in> B Co C |] ==> F \<in> A Co C"
   177 apply (simp add: Constrains_eq_constrains)
   178 apply (blast intro: constrains_trans constrains_weaken)
   179 done
   180 
   181 lemma Constrains_cancel:
   182      "[| F \<in> A Co (A' \<union> B); F \<in> B Co B' |] ==> F \<in> A Co (A' \<union> B')"
   183 apply (simp add: Constrains_eq_constrains constrains_def)
   184 apply best
   185 done
   186 
   187 
   188 subsection{*Stable*}
   189 
   190 (*Useful because there's no Stable_weaken.  [Tanja Vos]*)
   191 lemma Stable_eq: "[| F \<in> Stable A; A = B |] ==> F \<in> Stable B"
   192 by blast
   193 
   194 lemma Stable_eq_stable: "(F \<in> Stable A) = (F \<in> stable (reachable F \<inter> A))"
   195 by (simp add: Stable_def Constrains_eq_constrains stable_def)
   196 
   197 lemma StableI: "F \<in> A Co A ==> F \<in> Stable A"
   198 by (unfold Stable_def, assumption)
   199 
   200 lemma StableD: "F \<in> Stable A ==> F \<in> A Co A"
   201 by (unfold Stable_def, assumption)
   202 
   203 lemma Stable_Un: 
   204     "[| F \<in> Stable A; F \<in> Stable A' |] ==> F \<in> Stable (A \<union> A')"
   205 apply (unfold Stable_def)
   206 apply (blast intro: Constrains_Un)
   207 done
   208 
   209 lemma Stable_Int: 
   210     "[| F \<in> Stable A; F \<in> Stable A' |] ==> F \<in> Stable (A \<inter> A')"
   211 apply (unfold Stable_def)
   212 apply (blast intro: Constrains_Int)
   213 done
   214 
   215 lemma Stable_Constrains_Un: 
   216     "[| F \<in> Stable C; F \<in> A Co (C \<union> A') |]    
   217      ==> F \<in> (C \<union> A) Co (C \<union> A')"
   218 apply (unfold Stable_def)
   219 apply (blast intro: Constrains_Un [THEN Constrains_weaken])
   220 done
   221 
   222 lemma Stable_Constrains_Int: 
   223     "[| F \<in> Stable C; F \<in> (C \<inter> A) Co A' |]    
   224      ==> F \<in> (C \<inter> A) Co (C \<inter> A')"
   225 apply (unfold Stable_def)
   226 apply (blast intro: Constrains_Int [THEN Constrains_weaken])
   227 done
   228 
   229 lemma Stable_UN: 
   230     "(!!i. i \<in> I ==> F \<in> Stable (A i)) ==> F \<in> Stable (\<Union>i \<in> I. A i)"
   231 by (simp add: Stable_def Constrains_UN) 
   232 
   233 lemma Stable_INT: 
   234     "(!!i. i \<in> I ==> F \<in> Stable (A i)) ==> F \<in> Stable (\<Inter>i \<in> I. A i)"
   235 by (simp add: Stable_def Constrains_INT) 
   236 
   237 lemma Stable_reachable: "F \<in> Stable (reachable F)"
   238 by (simp add: Stable_eq_stable)
   239 
   240 
   241 
   242 subsection{*Increasing*}
   243 
   244 lemma IncreasingD: 
   245      "F \<in> Increasing f ==> F \<in> Stable {s. x \<le> f s}"
   246 by (unfold Increasing_def, blast)
   247 
   248 lemma mono_Increasing_o: 
   249      "mono g ==> Increasing f \<subseteq> Increasing (g o f)"
   250 apply (simp add: Increasing_def Stable_def Constrains_def stable_def 
   251                  constrains_def)
   252 apply (blast intro: monoD order_trans)
   253 done
   254 
   255 lemma strict_IncreasingD: 
   256      "!!z::nat. F \<in> Increasing f ==> F \<in> Stable {s. z < f s}"
   257 by (simp add: Increasing_def Suc_le_eq [symmetric])
   258 
   259 lemma increasing_imp_Increasing: 
   260      "F \<in> increasing f ==> F \<in> Increasing f"
   261 apply (unfold increasing_def Increasing_def)
   262 apply (blast intro: stable_imp_Stable)
   263 done
   264 
   265 lemmas Increasing_constant = increasing_constant [THEN increasing_imp_Increasing, iff]
   266 
   267 
   268 subsection{*The Elimination Theorem*}
   269 
   270 (*The "free" m has become universally quantified! Should the premise be !!m
   271 instead of \<forall>m ?  Would make it harder to use in forward proof.*)
   272 
   273 lemma Elimination: 
   274     "[| \<forall>m. F \<in> {s. s x = m} Co (B m) |]  
   275      ==> F \<in> {s. s x \<in> M} Co (\<Union>m \<in> M. B m)"
   276 by (unfold Constrains_def constrains_def, blast)
   277 
   278 (*As above, but for the trivial case of a one-variable state, in which the
   279   state is identified with its one variable.*)
   280 lemma Elimination_sing: 
   281     "(\<forall>m. F \<in> {m} Co (B m)) ==> F \<in> M Co (\<Union>m \<in> M. B m)"
   282 by (unfold Constrains_def constrains_def, blast)
   283 
   284 
   285 subsection{*Specialized laws for handling Always*}
   286 
   287 (** Natural deduction rules for "Always A" **)
   288 
   289 lemma AlwaysI: "[| Init F \<subseteq> A;  F \<in> Stable A |] ==> F \<in> Always A"
   290 by (simp add: Always_def)
   291 
   292 lemma AlwaysD: "F \<in> Always A ==> Init F \<subseteq> A & F \<in> Stable A"
   293 by (simp add: Always_def)
   294 
   295 lemmas AlwaysE = AlwaysD [THEN conjE]
   296 lemmas Always_imp_Stable = AlwaysD [THEN conjunct2]
   297 
   298 
   299 (*The set of all reachable states is Always*)
   300 lemma Always_includes_reachable: "F \<in> Always A ==> reachable F \<subseteq> A"
   301 apply (simp add: Stable_def Constrains_def constrains_def Always_def)
   302 apply (rule subsetI)
   303 apply (erule reachable.induct)
   304 apply (blast intro: reachable.intros)+
   305 done
   306 
   307 lemma invariant_imp_Always: 
   308      "F \<in> invariant A ==> F \<in> Always A"
   309 apply (unfold Always_def invariant_def Stable_def stable_def)
   310 apply (blast intro: constrains_imp_Constrains)
   311 done
   312 
   313 lemmas Always_reachable = invariant_reachable [THEN invariant_imp_Always]
   314 
   315 lemma Always_eq_invariant_reachable:
   316      "Always A = {F. F \<in> invariant (reachable F \<inter> A)}"
   317 apply (simp add: Always_def invariant_def Stable_def Constrains_eq_constrains
   318                  stable_def)
   319 apply (blast intro: reachable.intros)
   320 done
   321 
   322 (*the RHS is the traditional definition of the "always" operator*)
   323 lemma Always_eq_includes_reachable: "Always A = {F. reachable F \<subseteq> A}"
   324 by (auto dest: invariant_includes_reachable simp add: Int_absorb2 invariant_reachable Always_eq_invariant_reachable)
   325 
   326 lemma Always_UNIV_eq [simp]: "Always UNIV = UNIV"
   327 by (auto simp add: Always_eq_includes_reachable)
   328 
   329 lemma UNIV_AlwaysI: "UNIV \<subseteq> A ==> F \<in> Always A"
   330 by (auto simp add: Always_eq_includes_reachable)
   331 
   332 lemma Always_eq_UN_invariant: "Always A = (\<Union>I \<in> Pow A. invariant I)"
   333 apply (simp add: Always_eq_includes_reachable)
   334 apply (blast intro: invariantI Init_subset_reachable [THEN subsetD] 
   335                     invariant_includes_reachable [THEN subsetD])
   336 done
   337 
   338 lemma Always_weaken: "[| F \<in> Always A; A \<subseteq> B |] ==> F \<in> Always B"
   339 by (auto simp add: Always_eq_includes_reachable)
   340 
   341 
   342 subsection{*"Co" rules involving Always*}
   343 
   344 lemma Always_Constrains_pre:
   345      "F \<in> Always INV ==> (F \<in> (INV \<inter> A) Co A') = (F \<in> A Co A')"
   346 by (simp add: Always_includes_reachable [THEN Int_absorb2] Constrains_def 
   347               Int_assoc [symmetric])
   348 
   349 lemma Always_Constrains_post:
   350      "F \<in> Always INV ==> (F \<in> A Co (INV \<inter> A')) = (F \<in> A Co A')"
   351 by (simp add: Always_includes_reachable [THEN Int_absorb2] 
   352               Constrains_eq_constrains Int_assoc [symmetric])
   353 
   354 (* [| F \<in> Always INV;  F \<in> (INV \<inter> A) Co A' |] ==> F \<in> A Co A' *)
   355 lemmas Always_ConstrainsI = Always_Constrains_pre [THEN iffD1]
   356 
   357 (* [| F \<in> Always INV;  F \<in> A Co A' |] ==> F \<in> A Co (INV \<inter> A') *)
   358 lemmas Always_ConstrainsD = Always_Constrains_post [THEN iffD2]
   359 
   360 (*The analogous proof of Always_LeadsTo_weaken doesn't terminate*)
   361 lemma Always_Constrains_weaken:
   362      "[| F \<in> Always C;  F \<in> A Co A';    
   363          C \<inter> B \<subseteq> A;   C \<inter> A' \<subseteq> B' |]  
   364       ==> F \<in> B Co B'"
   365 apply (rule Always_ConstrainsI, assumption)
   366 apply (drule Always_ConstrainsD, assumption)
   367 apply (blast intro: Constrains_weaken)
   368 done
   369 
   370 
   371 (** Conjoining Always properties **)
   372 
   373 lemma Always_Int_distrib: "Always (A \<inter> B) = Always A \<inter> Always B"
   374 by (auto simp add: Always_eq_includes_reachable)
   375 
   376 lemma Always_INT_distrib: "Always (INTER I A) = (\<Inter>i \<in> I. Always (A i))"
   377 by (auto simp add: Always_eq_includes_reachable)
   378 
   379 lemma Always_Int_I:
   380      "[| F \<in> Always A;  F \<in> Always B |] ==> F \<in> Always (A \<inter> B)"
   381 by (simp add: Always_Int_distrib)
   382 
   383 (*Allows a kind of "implication introduction"*)
   384 lemma Always_Compl_Un_eq:
   385      "F \<in> Always A ==> (F \<in> Always (-A \<union> B)) = (F \<in> Always B)"
   386 by (auto simp add: Always_eq_includes_reachable)
   387 
   388 (*Delete the nearest invariance assumption (which will be the second one
   389   used by Always_Int_I) *)
   390 lemmas Always_thin = thin_rl [of "F \<in> Always A"]
   391 
   392 
   393 subsection{*Totalize*}
   394 
   395 lemma reachable_imp_reachable_tot:
   396       "s \<in> reachable F ==> s \<in> reachable (totalize F)"
   397 apply (erule reachable.induct)
   398  apply (rule reachable.Init) 
   399  apply simp 
   400 apply (rule_tac act = "totalize_act act" in reachable.Acts) 
   401 apply (auto simp add: totalize_act_def) 
   402 done
   403 
   404 lemma reachable_tot_imp_reachable:
   405       "s \<in> reachable (totalize F) ==> s \<in> reachable F"
   406 apply (erule reachable.induct)
   407  apply (rule reachable.Init, simp) 
   408 apply (force simp add: totalize_act_def intro: reachable.Acts) 
   409 done
   410 
   411 lemma reachable_tot_eq [simp]: "reachable (totalize F) = reachable F"
   412 by (blast intro: reachable_imp_reachable_tot reachable_tot_imp_reachable) 
   413 
   414 lemma totalize_Constrains_iff [simp]: "(totalize F \<in> A Co B) = (F \<in> A Co B)"
   415 by (simp add: Constrains_def) 
   416 
   417 lemma totalize_Stable_iff [simp]: "(totalize F \<in> Stable A) = (F \<in> Stable A)"
   418 by (simp add: Stable_def)
   419 
   420 lemma totalize_Always_iff [simp]: "(totalize F \<in> Always A) = (F \<in> Always A)"
   421 by (simp add: Always_def)
   422 
   423 end