src/HOL/UNITY/Constrains.thy
 changeset 13797 baefae13ad37 parent 6823 97babc436a41 child 13798 4c1a53627500
```     1.1 --- a/src/HOL/UNITY/Constrains.thy	Thu Jan 30 10:35:56 2003 +0100
1.2 +++ b/src/HOL/UNITY/Constrains.thy	Thu Jan 30 18:08:09 2003 +0100
1.3 @@ -3,10 +3,10 @@
1.4      Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
1.5      Copyright   1998  University of Cambridge
1.6
1.7 -Safety relations: restricted to the set of reachable states.
1.8 +Weak safety relations: restricted to the set of reachable states.
1.9  *)
1.10
1.11 -Constrains = UNITY +
1.12 +theory Constrains = UNITY:
1.13
1.14  consts traces :: "['a set, ('a * 'a)set set] => ('a * 'a list) set"
1.15
1.16 @@ -14,36 +14,30 @@
1.17      Arguments MUST be curried in an inductive definition*)
1.18
1.19  inductive "traces init acts"
1.20 -  intrs
1.21 +  intros
1.22           (*Initial trace is empty*)
1.23 -    Init  "s: init ==> (s,[]) : traces init acts"
1.24 +    Init:  "s: init ==> (s,[]) : traces init acts"
1.25
1.26 -    Acts  "[| act: acts;  (s,evs) : traces init acts;  (s,s'): act |]
1.27 -	   ==> (s', s#evs) : traces init acts"
1.28 +    Acts:  "[| act: acts;  (s,evs) : traces init acts;  (s,s'): act |]
1.29 +	    ==> (s', s#evs) : traces init acts"
1.30
1.31
1.32  consts reachable :: "'a program => 'a set"
1.33
1.34  inductive "reachable F"
1.35 -  intrs
1.36 -    Init  "s: Init F ==> s : reachable F"
1.37 -
1.38 -    Acts  "[| act: Acts F;  s : reachable F;  (s,s'): act |]
1.39 -	   ==> s' : reachable F"
1.40 +  intros
1.41 +    Init:  "s: Init F ==> s : reachable F"
1.42
1.43 -consts
1.44 -  Constrains :: "['a set, 'a set] => 'a program set"  (infixl "Co"     60)
1.45 -  op_Unless  :: "['a set, 'a set] => 'a program set"  (infixl "Unless" 60)
1.46 +    Acts:  "[| act: Acts F;  s : reachable F;  (s,s'): act |]
1.47 +	    ==> s' : reachable F"
1.48
1.49 -defs
1.50 -  Constrains_def
1.51 +constdefs
1.52 +  Constrains :: "['a set, 'a set] => 'a program set"  (infixl "Co" 60)
1.53      "A Co B == {F. F : (reachable F Int A)  co  B}"
1.54
1.55 -  Unless_def
1.56 +  Unless  :: "['a set, 'a set] => 'a program set"     (infixl "Unless" 60)
1.57      "A Unless B == (A-B) Co (A Un B)"
1.58
1.59 -constdefs
1.60 -
1.61    Stable     :: "'a set => 'a program set"
1.62      "Stable A == A Co A"
1.63
1.64 @@ -55,4 +49,345 @@
1.65    Increasing :: "['a => 'b::{order}] => 'a program set"
1.66      "Increasing f == INT z. Stable {s. z <= f s}"
1.67
1.68 +
1.69 +(*** traces and reachable ***)
1.70 +
1.71 +lemma reachable_equiv_traces:
1.72 +     "reachable F = {s. EX evs. (s,evs): traces (Init F) (Acts F)}"
1.73 +apply safe
1.74 +apply (erule_tac [2] traces.induct)
1.75 +apply (erule reachable.induct)
1.76 +apply (blast intro: reachable.intros traces.intros)+
1.77 +done
1.78 +
1.79 +lemma Init_subset_reachable: "Init F <= reachable F"
1.80 +by (blast intro: reachable.intros)
1.81 +
1.82 +lemma stable_reachable [intro!,simp]:
1.83 +     "Acts G <= Acts F ==> G : stable (reachable F)"
1.84 +by (blast intro: stableI constrainsI reachable.intros)
1.85 +
1.86 +(*The set of all reachable states is an invariant...*)
1.87 +lemma invariant_reachable: "F : invariant (reachable F)"
1.89 +apply (blast intro: reachable.intros)
1.90 +done
1.91 +
1.92 +(*...in fact the strongest invariant!*)
1.93 +lemma invariant_includes_reachable: "F : invariant A ==> reachable F <= A"
1.94 +apply (simp add: stable_def constrains_def invariant_def)
1.95 +apply (rule subsetI)
1.96 +apply (erule reachable.induct)
1.97 +apply (blast intro: reachable.intros)+
1.98 +done
1.99 +
1.100 +
1.101 +(*** Co ***)
1.102 +
1.103 +(*F : B co B' ==> F : (reachable F Int B) co (reachable F Int B')*)
1.104 +lemmas constrains_reachable_Int =
1.105 +    subset_refl [THEN stable_reachable [unfolded stable_def],
1.106 +                 THEN constrains_Int, standard]
1.107 +
1.108 +(*Resembles the previous definition of Constrains*)
1.109 +lemma Constrains_eq_constrains:
1.110 +     "A Co B = {F. F : (reachable F  Int  A) co (reachable F  Int  B)}"
1.111 +apply (unfold Constrains_def)
1.112 +apply (blast dest: constrains_reachable_Int intro: constrains_weaken)
1.113 +done
1.114 +
1.115 +lemma constrains_imp_Constrains: "F : A co A' ==> F : A Co A'"
1.116 +apply (unfold Constrains_def)
1.117 +apply (blast intro: constrains_weaken_L)
1.118 +done
1.119 +
1.120 +lemma stable_imp_Stable: "F : stable A ==> F : Stable A"
1.121 +apply (unfold stable_def Stable_def)
1.122 +apply (erule constrains_imp_Constrains)
1.123 +done
1.124 +
1.125 +lemma ConstrainsI:
1.126 +    "(!!act s s'. [| act: Acts F;  (s,s') : act;  s: A |] ==> s': A')
1.127 +     ==> F : A Co A'"
1.128 +apply (rule constrains_imp_Constrains)
1.129 +apply (blast intro: constrainsI)
1.130 +done
1.131 +
1.132 +lemma Constrains_empty [iff]: "F : {} Co B"
1.133 +by (unfold Constrains_def constrains_def, blast)
1.134 +
1.135 +lemma Constrains_UNIV [iff]: "F : A Co UNIV"
1.136 +by (blast intro: ConstrainsI)
1.137 +
1.138 +lemma Constrains_weaken_R:
1.139 +    "[| F : A Co A'; A'<=B' |] ==> F : A Co B'"
1.140 +apply (unfold Constrains_def)
1.141 +apply (blast intro: constrains_weaken_R)
1.142 +done
1.143 +
1.144 +lemma Constrains_weaken_L:
1.145 +    "[| F : A Co A'; B<=A |] ==> F : B Co A'"
1.146 +apply (unfold Constrains_def)
1.147 +apply (blast intro: constrains_weaken_L)
1.148 +done
1.149 +
1.150 +lemma Constrains_weaken:
1.151 +   "[| F : A Co A'; B<=A; A'<=B' |] ==> F : B Co B'"
1.152 +apply (unfold Constrains_def)
1.153 +apply (blast intro: constrains_weaken)
1.154 +done
1.155 +
1.156 +(** Union **)
1.157 +
1.158 +lemma Constrains_Un:
1.159 +    "[| F : A Co A'; F : B Co B' |] ==> F : (A Un B) Co (A' Un B')"
1.160 +apply (unfold Constrains_def)
1.161 +apply (blast intro: constrains_Un [THEN constrains_weaken])
1.162 +done
1.163 +
1.164 +lemma Constrains_UN:
1.165 +  assumes Co: "!!i. i:I ==> F : (A i) Co (A' i)"
1.166 +  shows "F : (UN i:I. A i) Co (UN i:I. A' i)"
1.167 +apply (unfold Constrains_def)
1.168 +apply (rule CollectI)
1.169 +apply (rule Co [unfolded Constrains_def, THEN CollectD, THEN constrains_UN,
1.170 +                THEN constrains_weaken],   auto)
1.171 +done
1.172 +
1.173 +(** Intersection **)
1.174 +
1.175 +lemma Constrains_Int:
1.176 +    "[| F : A Co A'; F : B Co B' |] ==> F : (A Int B) Co (A' Int B')"
1.177 +apply (unfold Constrains_def)
1.178 +apply (blast intro: constrains_Int [THEN constrains_weaken])
1.179 +done
1.180 +
1.181 +lemma Constrains_INT:
1.182 +  assumes Co: "!!i. i:I ==> F : (A i) Co (A' i)"
1.183 +  shows "F : (INT i:I. A i) Co (INT i:I. A' i)"
1.184 +apply (unfold Constrains_def)
1.185 +apply (rule CollectI)
1.186 +apply (rule Co [unfolded Constrains_def, THEN CollectD, THEN constrains_INT,
1.187 +                THEN constrains_weaken],   auto)
1.188 +done
1.189 +
1.190 +lemma Constrains_imp_subset: "F : A Co A' ==> reachable F Int A <= A'"
1.191 +by (simp add: constrains_imp_subset Constrains_def)
1.192 +
1.193 +lemma Constrains_trans: "[| F : A Co B; F : B Co C |] ==> F : A Co C"
1.195 +apply (blast intro: constrains_trans constrains_weaken)
1.196 +done
1.197 +
1.198 +lemma Constrains_cancel:
1.199 +     "[| F : A Co (A' Un B); F : B Co B' |] ==> F : A Co (A' Un B')"
1.200 +by (simp add: Constrains_eq_constrains constrains_def, blast)
1.201 +
1.202 +
1.203 +(*** Stable ***)
1.204 +
1.205 +(*Useful because there's no Stable_weaken.  [Tanja Vos]*)
1.206 +lemma Stable_eq: "[| F: Stable A; A = B |] ==> F : Stable B"
1.207 +by blast
1.208 +
1.209 +lemma Stable_eq_stable: "(F : Stable A) = (F : stable (reachable F Int A))"
1.210 +by (simp add: Stable_def Constrains_eq_constrains stable_def)
1.211 +
1.212 +lemma StableI: "F : A Co A ==> F : Stable A"
1.213 +by (unfold Stable_def, assumption)
1.214 +
1.215 +lemma StableD: "F : Stable A ==> F : A Co A"
1.216 +by (unfold Stable_def, assumption)
1.217 +
1.218 +lemma Stable_Un:
1.219 +    "[| F : Stable A; F : Stable A' |] ==> F : Stable (A Un A')"
1.220 +apply (unfold Stable_def)
1.221 +apply (blast intro: Constrains_Un)
1.222 +done
1.223 +
1.224 +lemma Stable_Int:
1.225 +    "[| F : Stable A; F : Stable A' |] ==> F : Stable (A Int A')"
1.226 +apply (unfold Stable_def)
1.227 +apply (blast intro: Constrains_Int)
1.228 +done
1.229 +
1.230 +lemma Stable_Constrains_Un:
1.231 +    "[| F : Stable C; F : A Co (C Un A') |]
1.232 +     ==> F : (C Un A) Co (C Un A')"
1.233 +apply (unfold Stable_def)
1.234 +apply (blast intro: Constrains_Un [THEN Constrains_weaken])
1.235 +done
1.236 +
1.237 +lemma Stable_Constrains_Int:
1.238 +    "[| F : Stable C; F : (C Int A) Co A' |]
1.239 +     ==> F : (C Int A) Co (C Int A')"
1.240 +apply (unfold Stable_def)
1.241 +apply (blast intro: Constrains_Int [THEN Constrains_weaken])
1.242 +done
1.243 +
1.244 +lemma Stable_UN:
1.245 +    "(!!i. i:I ==> F : Stable (A i)) ==> F : Stable (UN i:I. A i)"
1.246 +by (simp add: Stable_def Constrains_UN)
1.247 +
1.248 +lemma Stable_INT:
1.249 +    "(!!i. i:I ==> F : Stable (A i)) ==> F : Stable (INT i:I. A i)"
1.250 +by (simp add: Stable_def Constrains_INT)
1.251 +
1.252 +lemma Stable_reachable: "F : Stable (reachable F)"
1.254 +
1.255 +
1.256 +
1.257 +(*** Increasing ***)
1.258 +
1.259 +lemma IncreasingD:
1.260 +     "F : Increasing f ==> F : Stable {s. x <= f s}"
1.261 +by (unfold Increasing_def, blast)
1.262 +
1.263 +lemma mono_Increasing_o:
1.264 +     "mono g ==> Increasing f <= Increasing (g o f)"
1.265 +apply (simp add: Increasing_def Stable_def Constrains_def stable_def
1.266 +                 constrains_def)
1.267 +apply (blast intro: monoD order_trans)
1.268 +done
1.269 +
1.270 +lemma strict_IncreasingD:
1.271 +     "!!z::nat. F : Increasing f ==> F: Stable {s. z < f s}"
1.272 +by (simp add: Increasing_def Suc_le_eq [symmetric])
1.273 +
1.274 +lemma increasing_imp_Increasing:
1.275 +     "F : increasing f ==> F : Increasing f"
1.276 +apply (unfold increasing_def Increasing_def)
1.277 +apply (blast intro: stable_imp_Stable)
1.278 +done
1.279 +
1.280 +lemmas Increasing_constant =
1.281 +    increasing_constant [THEN increasing_imp_Increasing, standard, iff]
1.282 +
1.283 +
1.284 +(*** The Elimination Theorem.  The "free" m has become universally quantified!
1.285 +     Should the premise be !!m instead of ALL m ?  Would make it harder to use
1.286 +     in forward proof. ***)
1.287 +
1.288 +lemma Elimination:
1.289 +    "[| ALL m. F : {s. s x = m} Co (B m) |]
1.290 +     ==> F : {s. s x : M} Co (UN m:M. B m)"
1.291 +
1.292 +by (unfold Constrains_def constrains_def, blast)
1.293 +
1.294 +(*As above, but for the trivial case of a one-variable state, in which the
1.295 +  state is identified with its one variable.*)
1.296 +lemma Elimination_sing:
1.297 +    "(ALL m. F : {m} Co (B m)) ==> F : M Co (UN m:M. B m)"
1.298 +by (unfold Constrains_def constrains_def, blast)
1.299 +
1.300 +
1.301 +(*** Specialized laws for handling Always ***)
1.302 +
1.303 +(** Natural deduction rules for "Always A" **)
1.304 +
1.305 +lemma AlwaysI: "[| Init F<=A;  F : Stable A |] ==> F : Always A"
1.307 +
1.308 +lemma AlwaysD: "F : Always A ==> Init F<=A & F : Stable A"
1.310 +
1.311 +lemmas AlwaysE = AlwaysD [THEN conjE, standard]
1.312 +lemmas Always_imp_Stable = AlwaysD [THEN conjunct2, standard]
1.313 +
1.314 +
1.315 +(*The set of all reachable states is Always*)
1.316 +lemma Always_includes_reachable: "F : Always A ==> reachable F <= A"
1.317 +apply (simp add: Stable_def Constrains_def constrains_def Always_def)
1.318 +apply (rule subsetI)
1.319 +apply (erule reachable.induct)
1.320 +apply (blast intro: reachable.intros)+
1.321 +done
1.322 +
1.323 +lemma invariant_imp_Always:
1.324 +     "F : invariant A ==> F : Always A"
1.325 +apply (unfold Always_def invariant_def Stable_def stable_def)
1.326 +apply (blast intro: constrains_imp_Constrains)
1.327 +done
1.328 +
1.329 +lemmas Always_reachable =
1.330 +    invariant_reachable [THEN invariant_imp_Always, standard]
1.331 +
1.332 +lemma Always_eq_invariant_reachable:
1.333 +     "Always A = {F. F : invariant (reachable F Int A)}"
1.334 +apply (simp add: Always_def invariant_def Stable_def Constrains_eq_constrains
1.335 +                 stable_def)
1.336 +apply (blast intro: reachable.intros)
1.337 +done
1.338 +
1.339 +(*the RHS is the traditional definition of the "always" operator*)
1.340 +lemma Always_eq_includes_reachable: "Always A = {F. reachable F <= A}"
1.341 +by (auto dest: invariant_includes_reachable simp add: Int_absorb2 invariant_reachable Always_eq_invariant_reachable)
1.342 +
1.343 +lemma Always_UNIV_eq [simp]: "Always UNIV = UNIV"
1.344 +by (auto simp add: Always_eq_includes_reachable)
1.345 +
1.346 +lemma UNIV_AlwaysI: "UNIV <= A ==> F : Always A"
1.347 +by (auto simp add: Always_eq_includes_reachable)
1.348 +
1.349 +lemma Always_eq_UN_invariant: "Always A = (UN I: Pow A. invariant I)"
1.351 +apply (blast intro: invariantI Init_subset_reachable [THEN subsetD]
1.352 +                    invariant_includes_reachable [THEN subsetD])
1.353 +done
1.354 +
1.355 +lemma Always_weaken: "[| F : Always A; A <= B |] ==> F : Always B"
1.356 +by (auto simp add: Always_eq_includes_reachable)
1.357 +
1.358 +
1.359 +(*** "Co" rules involving Always ***)
1.360 +
1.361 +lemma Always_Constrains_pre:
1.362 +     "F : Always INV ==> (F : (INV Int A) Co A') = (F : A Co A')"
1.363 +by (simp add: Always_includes_reachable [THEN Int_absorb2] Constrains_def
1.364 +              Int_assoc [symmetric])
1.365 +
1.366 +lemma Always_Constrains_post:
1.367 +     "F : Always INV ==> (F : A Co (INV Int A')) = (F : A Co A')"
1.368 +by (simp add: Always_includes_reachable [THEN Int_absorb2]
1.369 +              Constrains_eq_constrains Int_assoc [symmetric])
1.370 +
1.371 +(* [| F : Always INV;  F : (INV Int A) Co A' |] ==> F : A Co A' *)
1.372 +lemmas Always_ConstrainsI = Always_Constrains_pre [THEN iffD1, standard]
1.373 +
1.374 +(* [| F : Always INV;  F : A Co A' |] ==> F : A Co (INV Int A') *)
1.375 +lemmas Always_ConstrainsD = Always_Constrains_post [THEN iffD2, standard]
1.376 +
1.377 +(*The analogous proof of Always_LeadsTo_weaken doesn't terminate*)
1.378 +lemma Always_Constrains_weaken:
1.379 +     "[| F : Always C;  F : A Co A';
1.380 +         C Int B <= A;   C Int A' <= B' |]
1.381 +      ==> F : B Co B'"
1.382 +apply (rule Always_ConstrainsI, assumption)
1.383 +apply (drule Always_ConstrainsD, assumption)
1.384 +apply (blast intro: Constrains_weaken)
1.385 +done
1.386 +
1.387 +
1.388 +(** Conjoining Always properties **)
1.389 +
1.390 +lemma Always_Int_distrib: "Always (A Int B) = Always A Int Always B"
1.391 +by (auto simp add: Always_eq_includes_reachable)
1.392 +
1.393 +lemma Always_INT_distrib: "Always (INTER I A) = (INT i:I. Always (A i))"
1.394 +by (auto simp add: Always_eq_includes_reachable)
1.395 +
1.396 +lemma Always_Int_I:
1.397 +     "[| F : Always A;  F : Always B |] ==> F : Always (A Int B)"