src/HOL/UNITY/SubstAx.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 62343 24106dc44def child 63146 f1ecba0272f9 permissions -rw-r--r--
Lots of new material for multivariate analysis
```     1 (*  Title:      HOL/UNITY/SubstAx.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1998  University of Cambridge
```
```     4
```
```     5 Weak LeadsTo relation (restricted to the set of reachable states)
```
```     6 *)
```
```     7
```
```     8 section{*Weak Progress*}
```
```     9
```
```    10 theory SubstAx imports WFair Constrains begin
```
```    11
```
```    12 definition Ensures :: "['a set, 'a set] => 'a program set" (infixl "Ensures" 60) where
```
```    13     "A Ensures B == {F. F \<in> (reachable F \<inter> A) ensures B}"
```
```    14
```
```    15 definition LeadsTo :: "['a set, 'a set] => 'a program set" (infixl "LeadsTo" 60) where
```
```    16     "A LeadsTo B == {F. F \<in> (reachable F \<inter> A) leadsTo B}"
```
```    17
```
```    18 notation LeadsTo  (infixl "\<longmapsto>w" 60)
```
```    19
```
```    20
```
```    21 text{*Resembles the previous definition of LeadsTo*}
```
```    22 lemma LeadsTo_eq_leadsTo:
```
```    23      "A LeadsTo B = {F. F \<in> (reachable F \<inter> A) leadsTo (reachable F \<inter> B)}"
```
```    24 apply (unfold LeadsTo_def)
```
```    25 apply (blast dest: psp_stable2 intro: leadsTo_weaken)
```
```    26 done
```
```    27
```
```    28
```
```    29 subsection{*Specialized laws for handling invariants*}
```
```    30
```
```    31 (** Conjoining an Always property **)
```
```    32
```
```    33 lemma Always_LeadsTo_pre:
```
```    34      "F \<in> Always INV ==> (F \<in> (INV \<inter> A) LeadsTo A') = (F \<in> A LeadsTo A')"
```
```    35 by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2
```
```    36               Int_assoc [symmetric])
```
```    37
```
```    38 lemma Always_LeadsTo_post:
```
```    39      "F \<in> Always INV ==> (F \<in> A LeadsTo (INV \<inter> A')) = (F \<in> A LeadsTo A')"
```
```    40 by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2
```
```    41               Int_assoc [symmetric])
```
```    42
```
```    43 (* [| F \<in> Always C;  F \<in> (C \<inter> A) LeadsTo A' |] ==> F \<in> A LeadsTo A' *)
```
```    44 lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1]
```
```    45
```
```    46 (* [| F \<in> Always INV;  F \<in> A LeadsTo A' |] ==> F \<in> A LeadsTo (INV \<inter> A') *)
```
```    47 lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2]
```
```    48
```
```    49
```
```    50 subsection{*Introduction rules: Basis, Trans, Union*}
```
```    51
```
```    52 lemma leadsTo_imp_LeadsTo: "F \<in> A leadsTo B ==> F \<in> A LeadsTo B"
```
```    53 apply (simp add: LeadsTo_def)
```
```    54 apply (blast intro: leadsTo_weaken_L)
```
```    55 done
```
```    56
```
```    57 lemma LeadsTo_Trans:
```
```    58      "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |] ==> F \<in> A LeadsTo C"
```
```    59 apply (simp add: LeadsTo_eq_leadsTo)
```
```    60 apply (blast intro: leadsTo_Trans)
```
```    61 done
```
```    62
```
```    63 lemma LeadsTo_Union:
```
```    64      "(!!A. A \<in> S ==> F \<in> A LeadsTo B) ==> F \<in> (\<Union>S) LeadsTo B"
```
```    65 apply (simp add: LeadsTo_def)
```
```    66 apply (subst Int_Union)
```
```    67 apply (blast intro: leadsTo_UN)
```
```    68 done
```
```    69
```
```    70
```
```    71 subsection{*Derived rules*}
```
```    72
```
```    73 lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"
```
```    74 by (simp add: LeadsTo_def)
```
```    75
```
```    76 text{*Useful with cancellation, disjunction*}
```
```    77 lemma LeadsTo_Un_duplicate:
```
```    78      "F \<in> A LeadsTo (A' \<union> A') ==> F \<in> A LeadsTo A'"
```
```    79 by (simp add: Un_ac)
```
```    80
```
```    81 lemma LeadsTo_Un_duplicate2:
```
```    82      "F \<in> A LeadsTo (A' \<union> C \<union> C) ==> F \<in> A LeadsTo (A' \<union> C)"
```
```    83 by (simp add: Un_ac)
```
```    84
```
```    85 lemma LeadsTo_UN:
```
```    86      "(!!i. i \<in> I ==> F \<in> (A i) LeadsTo B) ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo B"
```
```    87 apply (blast intro: LeadsTo_Union)
```
```    88 done
```
```    89
```
```    90 text{*Binary union introduction rule*}
```
```    91 lemma LeadsTo_Un:
```
```    92      "[| F \<in> A LeadsTo C; F \<in> B LeadsTo C |] ==> F \<in> (A \<union> B) LeadsTo C"
```
```    93   using LeadsTo_UN [of "{A, B}" F id C] by auto
```
```    94
```
```    95 text{*Lets us look at the starting state*}
```
```    96 lemma single_LeadsTo_I:
```
```    97      "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"
```
```    98 by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)
```
```    99
```
```   100 lemma subset_imp_LeadsTo: "A \<subseteq> B ==> F \<in> A LeadsTo B"
```
```   101 apply (simp add: LeadsTo_def)
```
```   102 apply (blast intro: subset_imp_leadsTo)
```
```   103 done
```
```   104
```
```   105 lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, simp]
```
```   106
```
```   107 lemma LeadsTo_weaken_R:
```
```   108      "[| F \<in> A LeadsTo A';  A' \<subseteq> B' |] ==> F \<in> A LeadsTo B'"
```
```   109 apply (simp add: LeadsTo_def)
```
```   110 apply (blast intro: leadsTo_weaken_R)
```
```   111 done
```
```   112
```
```   113 lemma LeadsTo_weaken_L:
```
```   114      "[| F \<in> A LeadsTo A';  B \<subseteq> A |]
```
```   115       ==> F \<in> B LeadsTo A'"
```
```   116 apply (simp add: LeadsTo_def)
```
```   117 apply (blast intro: leadsTo_weaken_L)
```
```   118 done
```
```   119
```
```   120 lemma LeadsTo_weaken:
```
```   121      "[| F \<in> A LeadsTo A';
```
```   122          B  \<subseteq> A;   A' \<subseteq> B' |]
```
```   123       ==> F \<in> B LeadsTo B'"
```
```   124 by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)
```
```   125
```
```   126 lemma Always_LeadsTo_weaken:
```
```   127      "[| F \<in> Always C;  F \<in> A LeadsTo A';
```
```   128          C \<inter> B \<subseteq> A;   C \<inter> A' \<subseteq> B' |]
```
```   129       ==> F \<in> B LeadsTo B'"
```
```   130 by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)
```
```   131
```
```   132 (** Two theorems for "proof lattices" **)
```
```   133
```
```   134 lemma LeadsTo_Un_post: "F \<in> A LeadsTo B ==> F \<in> (A \<union> B) LeadsTo B"
```
```   135 by (blast intro: LeadsTo_Un subset_imp_LeadsTo)
```
```   136
```
```   137 lemma LeadsTo_Trans_Un:
```
```   138      "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |]
```
```   139       ==> F \<in> (A \<union> B) LeadsTo C"
```
```   140 by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)
```
```   141
```
```   142
```
```   143 (** Distributive laws **)
```
```   144
```
```   145 lemma LeadsTo_Un_distrib:
```
```   146      "(F \<in> (A \<union> B) LeadsTo C)  = (F \<in> A LeadsTo C & F \<in> B LeadsTo C)"
```
```   147 by (blast intro: LeadsTo_Un LeadsTo_weaken_L)
```
```   148
```
```   149 lemma LeadsTo_UN_distrib:
```
```   150      "(F \<in> (\<Union>i \<in> I. A i) LeadsTo B)  =  (\<forall>i \<in> I. F \<in> (A i) LeadsTo B)"
```
```   151 by (blast intro: LeadsTo_UN LeadsTo_weaken_L)
```
```   152
```
```   153 lemma LeadsTo_Union_distrib:
```
```   154      "(F \<in> (\<Union>S) LeadsTo B)  =  (\<forall>A \<in> S. F \<in> A LeadsTo B)"
```
```   155 by (blast intro: LeadsTo_Union LeadsTo_weaken_L)
```
```   156
```
```   157
```
```   158 (** More rules using the premise "Always INV" **)
```
```   159
```
```   160 lemma LeadsTo_Basis: "F \<in> A Ensures B ==> F \<in> A LeadsTo B"
```
```   161 by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)
```
```   162
```
```   163 lemma EnsuresI:
```
```   164      "[| F \<in> (A-B) Co (A \<union> B);  F \<in> transient (A-B) |]
```
```   165       ==> F \<in> A Ensures B"
```
```   166 apply (simp add: Ensures_def Constrains_eq_constrains)
```
```   167 apply (blast intro: ensuresI constrains_weaken transient_strengthen)
```
```   168 done
```
```   169
```
```   170 lemma Always_LeadsTo_Basis:
```
```   171      "[| F \<in> Always INV;
```
```   172          F \<in> (INV \<inter> (A-A')) Co (A \<union> A');
```
```   173          F \<in> transient (INV \<inter> (A-A')) |]
```
```   174   ==> F \<in> A LeadsTo A'"
```
```   175 apply (rule Always_LeadsToI, assumption)
```
```   176 apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
```
```   177 done
```
```   178
```
```   179 text{*Set difference: maybe combine with @{text leadsTo_weaken_L}??
```
```   180   This is the most useful form of the "disjunction" rule*}
```
```   181 lemma LeadsTo_Diff:
```
```   182      "[| F \<in> (A-B) LeadsTo C;  F \<in> (A \<inter> B) LeadsTo C |]
```
```   183       ==> F \<in> A LeadsTo C"
```
```   184 by (blast intro: LeadsTo_Un LeadsTo_weaken)
```
```   185
```
```   186
```
```   187 lemma LeadsTo_UN_UN:
```
```   188      "(!! i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i))
```
```   189       ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"
```
```   190 apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
```
```   191 done
```
```   192
```
```   193
```
```   194 text{*Version with no index set*}
```
```   195 lemma LeadsTo_UN_UN_noindex:
```
```   196      "(!!i. F \<in> (A i) LeadsTo (A' i)) ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   197 by (blast intro: LeadsTo_UN_UN)
```
```   198
```
```   199 text{*Version with no index set*}
```
```   200 lemma all_LeadsTo_UN_UN:
```
```   201      "\<forall>i. F \<in> (A i) LeadsTo (A' i)
```
```   202       ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   203 by (blast intro: LeadsTo_UN_UN)
```
```   204
```
```   205 text{*Binary union version*}
```
```   206 lemma LeadsTo_Un_Un:
```
```   207      "[| F \<in> A LeadsTo A'; F \<in> B LeadsTo B' |]
```
```   208             ==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"
```
```   209 by (blast intro: LeadsTo_Un LeadsTo_weaken_R)
```
```   210
```
```   211
```
```   212 (** The cancellation law **)
```
```   213
```
```   214 lemma LeadsTo_cancel2:
```
```   215      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> B LeadsTo B' |]
```
```   216       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   217 by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)
```
```   218
```
```   219 lemma LeadsTo_cancel_Diff2:
```
```   220      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> (B-A') LeadsTo B' |]
```
```   221       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   222 apply (rule LeadsTo_cancel2)
```
```   223 prefer 2 apply assumption
```
```   224 apply (simp_all (no_asm_simp))
```
```   225 done
```
```   226
```
```   227 lemma LeadsTo_cancel1:
```
```   228      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> B LeadsTo B' |]
```
```   229       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   230 apply (simp add: Un_commute)
```
```   231 apply (blast intro!: LeadsTo_cancel2)
```
```   232 done
```
```   233
```
```   234 lemma LeadsTo_cancel_Diff1:
```
```   235      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> (B-A') LeadsTo B' |]
```
```   236       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   237 apply (rule LeadsTo_cancel1)
```
```   238 prefer 2 apply assumption
```
```   239 apply (simp_all (no_asm_simp))
```
```   240 done
```
```   241
```
```   242
```
```   243 text{*The impossibility law*}
```
```   244
```
```   245 text{*The set "A" may be non-empty, but it contains no reachable states*}
```
```   246 lemma LeadsTo_empty: "[|F \<in> A LeadsTo {}; all_total F|] ==> F \<in> Always (-A)"
```
```   247 apply (simp add: LeadsTo_def Always_eq_includes_reachable)
```
```   248 apply (drule leadsTo_empty, auto)
```
```   249 done
```
```   250
```
```   251
```
```   252 subsection{*PSP: Progress-Safety-Progress*}
```
```   253
```
```   254 text{*Special case of PSP: Misra's "stable conjunction"*}
```
```   255 lemma PSP_Stable:
```
```   256      "[| F \<in> A LeadsTo A';  F \<in> Stable B |]
```
```   257       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"
```
```   258 apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
```
```   259 apply (drule psp_stable, assumption)
```
```   260 apply (simp add: Int_ac)
```
```   261 done
```
```   262
```
```   263 lemma PSP_Stable2:
```
```   264      "[| F \<in> A LeadsTo A'; F \<in> Stable B |]
```
```   265       ==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"
```
```   266 by (simp add: PSP_Stable Int_ac)
```
```   267
```
```   268 lemma PSP:
```
```   269      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   270       ==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"
```
```   271 apply (simp add: LeadsTo_def Constrains_eq_constrains)
```
```   272 apply (blast dest: psp intro: leadsTo_weaken)
```
```   273 done
```
```   274
```
```   275 lemma PSP2:
```
```   276      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   277       ==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"
```
```   278 by (simp add: PSP Int_ac)
```
```   279
```
```   280 lemma PSP_Unless:
```
```   281      "[| F \<in> A LeadsTo A'; F \<in> B Unless B' |]
```
```   282       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"
```
```   283 apply (unfold Unless_def)
```
```   284 apply (drule PSP, assumption)
```
```   285 apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
```
```   286 done
```
```   287
```
```   288
```
```   289 lemma Stable_transient_Always_LeadsTo:
```
```   290      "[| F \<in> Stable A;  F \<in> transient C;
```
```   291          F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"
```
```   292 apply (erule Always_LeadsTo_weaken)
```
```   293 apply (rule LeadsTo_Diff)
```
```   294    prefer 2
```
```   295    apply (erule
```
```   296           transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])
```
```   297    apply (blast intro: subset_imp_LeadsTo)+
```
```   298 done
```
```   299
```
```   300
```
```   301 subsection{*Induction rules*}
```
```   302
```
```   303 (** Meta or object quantifier ????? **)
```
```   304 lemma LeadsTo_wf_induct:
```
```   305      "[| wf r;
```
```   306          \<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   307                     ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   308       ==> F \<in> A LeadsTo B"
```
```   309 apply (simp add: LeadsTo_eq_leadsTo)
```
```   310 apply (erule leadsTo_wf_induct)
```
```   311 apply (blast intro: leadsTo_weaken)
```
```   312 done
```
```   313
```
```   314
```
```   315 lemma Bounded_induct:
```
```   316      "[| wf r;
```
```   317          \<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   318                       ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   319       ==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"
```
```   320 apply (erule LeadsTo_wf_induct, safe)
```
```   321 apply (case_tac "m \<in> I")
```
```   322 apply (blast intro: LeadsTo_weaken)
```
```   323 apply (blast intro: subset_imp_LeadsTo)
```
```   324 done
```
```   325
```
```   326
```
```   327 lemma LessThan_induct:
```
```   328      "(!!m::nat. F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B))
```
```   329       ==> F \<in> A LeadsTo B"
```
```   330 by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
```
```   331
```
```   332 text{*Integer version.  Could generalize from 0 to any lower bound*}
```
```   333 lemma integ_0_le_induct:
```
```   334      "[| F \<in> Always {s. (0::int) \<le> f s};
```
```   335          !! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo
```
```   336                    ((A \<inter> {s. f s < z}) \<union> B) |]
```
```   337       ==> F \<in> A LeadsTo B"
```
```   338 apply (rule_tac f = "nat o f" in LessThan_induct)
```
```   339 apply (simp add: vimage_def)
```
```   340 apply (rule Always_LeadsTo_weaken, assumption+)
```
```   341 apply (auto simp add: nat_eq_iff nat_less_iff)
```
```   342 done
```
```   343
```
```   344 lemma LessThan_bounded_induct:
```
```   345      "!!l::nat. \<forall>m \<in> greaterThan l.
```
```   346                    F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)
```
```   347             ==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"
```
```   348 apply (simp only: Diff_eq [symmetric] vimage_Compl
```
```   349                   Compl_greaterThan [symmetric])
```
```   350 apply (rule wf_less_than [THEN Bounded_induct], simp)
```
```   351 done
```
```   352
```
```   353 lemma GreaterThan_bounded_induct:
```
```   354      "!!l::nat. \<forall>m \<in> lessThan l.
```
```   355                  F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)
```
```   356       ==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"
```
```   357 apply (rule_tac f = f and f1 = "%k. l - k"
```
```   358        in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
```
```   359 apply (simp add: Image_singleton, clarify)
```
```   360 apply (case_tac "m<l")
```
```   361  apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
```
```   362 apply (blast intro: not_le_imp_less subset_imp_LeadsTo)
```
```   363 done
```
```   364
```
```   365
```
```   366 subsection{*Completion: Binary and General Finite versions*}
```
```   367
```
```   368 lemma Completion:
```
```   369      "[| F \<in> A LeadsTo (A' \<union> C);  F \<in> A' Co (A' \<union> C);
```
```   370          F \<in> B LeadsTo (B' \<union> C);  F \<in> B' Co (B' \<union> C) |]
```
```   371       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
```
```   372 apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
```
```   373 apply (blast intro: completion leadsTo_weaken)
```
```   374 done
```
```   375
```
```   376 lemma Finite_completion_lemma:
```
```   377      "finite I
```
```   378       ==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->
```
```   379           (\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->
```
```   380           F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   381 apply (erule finite_induct, auto)
```
```   382 apply (rule Completion)
```
```   383    prefer 4
```
```   384    apply (simp only: INT_simps [symmetric])
```
```   385    apply (rule Constrains_INT, auto)
```
```   386 done
```
```   387
```
```   388 lemma Finite_completion:
```
```   389      "[| finite I;
```
```   390          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);
```
```   391          !!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]
```
```   392       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   393 by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])
```
```   394
```
```   395 lemma Stable_completion:
```
```   396      "[| F \<in> A LeadsTo A';  F \<in> Stable A';
```
```   397          F \<in> B LeadsTo B';  F \<in> Stable B' |]
```
```   398       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"
```
```   399 apply (unfold Stable_def)
```
```   400 apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
```
```   401 apply (force+)
```
```   402 done
```
```   403
```
```   404 lemma Finite_stable_completion:
```
```   405      "[| finite I;
```
```   406          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);
```
```   407          !!i. i \<in> I ==> F \<in> Stable (A' i) |]
```
```   408       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"
```
```   409 apply (unfold Stable_def)
```
```   410 apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
```
```   411 apply (simp_all, blast+)
```
```   412 done
```
```   413
```
```   414 end
```