src/HOL/UNITY/SubstAx.thy
 author hoelzl Thu Sep 02 10:14:32 2010 +0200 (2010-09-02) changeset 39072 1030b1a166ef parent 37936 1e4c5015a72e child 44106 0e018cbcc0de permissions -rw-r--r--
```     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 header{*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 (xsymbols)
```
```    19   LeadsTo  (infixl " \<longmapsto>w " 60)
```
```    20
```
```    21
```
```    22 text{*Resembles the previous definition of LeadsTo*}
```
```    23 lemma LeadsTo_eq_leadsTo:
```
```    24      "A LeadsTo B = {F. F \<in> (reachable F \<inter> A) leadsTo (reachable F \<inter> B)}"
```
```    25 apply (unfold LeadsTo_def)
```
```    26 apply (blast dest: psp_stable2 intro: leadsTo_weaken)
```
```    27 done
```
```    28
```
```    29
```
```    30 subsection{*Specialized laws for handling invariants*}
```
```    31
```
```    32 (** Conjoining an Always property **)
```
```    33
```
```    34 lemma Always_LeadsTo_pre:
```
```    35      "F \<in> Always INV ==> (F \<in> (INV \<inter> A) LeadsTo A') = (F \<in> A LeadsTo A')"
```
```    36 by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2
```
```    37               Int_assoc [symmetric])
```
```    38
```
```    39 lemma Always_LeadsTo_post:
```
```    40      "F \<in> Always INV ==> (F \<in> A LeadsTo (INV \<inter> A')) = (F \<in> A LeadsTo A')"
```
```    41 by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2
```
```    42               Int_assoc [symmetric])
```
```    43
```
```    44 (* [| F \<in> Always C;  F \<in> (C \<inter> A) LeadsTo A' |] ==> F \<in> A LeadsTo A' *)
```
```    45 lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1, standard]
```
```    46
```
```    47 (* [| F \<in> Always INV;  F \<in> A LeadsTo A' |] ==> F \<in> A LeadsTo (INV \<inter> A') *)
```
```    48 lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2, standard]
```
```    49
```
```    50
```
```    51 subsection{*Introduction rules: Basis, Trans, Union*}
```
```    52
```
```    53 lemma leadsTo_imp_LeadsTo: "F \<in> A leadsTo B ==> F \<in> A LeadsTo B"
```
```    54 apply (simp add: LeadsTo_def)
```
```    55 apply (blast intro: leadsTo_weaken_L)
```
```    56 done
```
```    57
```
```    58 lemma LeadsTo_Trans:
```
```    59      "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |] ==> F \<in> A LeadsTo C"
```
```    60 apply (simp add: LeadsTo_eq_leadsTo)
```
```    61 apply (blast intro: leadsTo_Trans)
```
```    62 done
```
```    63
```
```    64 lemma LeadsTo_Union:
```
```    65      "(!!A. A \<in> S ==> F \<in> A LeadsTo B) ==> F \<in> (Union S) LeadsTo B"
```
```    66 apply (simp add: LeadsTo_def)
```
```    67 apply (subst Int_Union)
```
```    68 apply (blast intro: leadsTo_UN)
```
```    69 done
```
```    70
```
```    71
```
```    72 subsection{*Derived rules*}
```
```    73
```
```    74 lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"
```
```    75 by (simp add: LeadsTo_def)
```
```    76
```
```    77 text{*Useful with cancellation, disjunction*}
```
```    78 lemma LeadsTo_Un_duplicate:
```
```    79      "F \<in> A LeadsTo (A' \<union> A') ==> F \<in> A LeadsTo A'"
```
```    80 by (simp add: Un_ac)
```
```    81
```
```    82 lemma LeadsTo_Un_duplicate2:
```
```    83      "F \<in> A LeadsTo (A' \<union> C \<union> C) ==> F \<in> A LeadsTo (A' \<union> C)"
```
```    84 by (simp add: Un_ac)
```
```    85
```
```    86 lemma LeadsTo_UN:
```
```    87      "(!!i. i \<in> I ==> F \<in> (A i) LeadsTo B) ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo B"
```
```    88 apply (simp only: Union_image_eq [symmetric])
```
```    89 apply (blast intro: LeadsTo_Union)
```
```    90 done
```
```    91
```
```    92 text{*Binary union introduction rule*}
```
```    93 lemma LeadsTo_Un:
```
```    94      "[| F \<in> A LeadsTo C; F \<in> B LeadsTo C |] ==> F \<in> (A \<union> B) LeadsTo C"
```
```    95 apply (subst Un_eq_Union)
```
```    96 apply (blast intro: LeadsTo_Union)
```
```    97 done
```
```    98
```
```    99 text{*Lets us look at the starting state*}
```
```   100 lemma single_LeadsTo_I:
```
```   101      "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"
```
```   102 by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)
```
```   103
```
```   104 lemma subset_imp_LeadsTo: "A \<subseteq> B ==> F \<in> A LeadsTo B"
```
```   105 apply (simp add: LeadsTo_def)
```
```   106 apply (blast intro: subset_imp_leadsTo)
```
```   107 done
```
```   108
```
```   109 lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, standard, simp]
```
```   110
```
```   111 lemma LeadsTo_weaken_R [rule_format]:
```
```   112      "[| F \<in> A LeadsTo A';  A' \<subseteq> B' |] ==> F \<in> A LeadsTo B'"
```
```   113 apply (simp add: LeadsTo_def)
```
```   114 apply (blast intro: leadsTo_weaken_R)
```
```   115 done
```
```   116
```
```   117 lemma LeadsTo_weaken_L [rule_format]:
```
```   118      "[| F \<in> A LeadsTo A';  B \<subseteq> A |]
```
```   119       ==> F \<in> B LeadsTo A'"
```
```   120 apply (simp add: LeadsTo_def)
```
```   121 apply (blast intro: leadsTo_weaken_L)
```
```   122 done
```
```   123
```
```   124 lemma LeadsTo_weaken:
```
```   125      "[| F \<in> A LeadsTo A';
```
```   126          B  \<subseteq> A;   A' \<subseteq> B' |]
```
```   127       ==> F \<in> B LeadsTo B'"
```
```   128 by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)
```
```   129
```
```   130 lemma Always_LeadsTo_weaken:
```
```   131      "[| F \<in> Always C;  F \<in> A LeadsTo A';
```
```   132          C \<inter> B \<subseteq> A;   C \<inter> A' \<subseteq> B' |]
```
```   133       ==> F \<in> B LeadsTo B'"
```
```   134 by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)
```
```   135
```
```   136 (** Two theorems for "proof lattices" **)
```
```   137
```
```   138 lemma LeadsTo_Un_post: "F \<in> A LeadsTo B ==> F \<in> (A \<union> B) LeadsTo B"
```
```   139 by (blast intro: LeadsTo_Un subset_imp_LeadsTo)
```
```   140
```
```   141 lemma LeadsTo_Trans_Un:
```
```   142      "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |]
```
```   143       ==> F \<in> (A \<union> B) LeadsTo C"
```
```   144 by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)
```
```   145
```
```   146
```
```   147 (** Distributive laws **)
```
```   148
```
```   149 lemma LeadsTo_Un_distrib:
```
```   150      "(F \<in> (A \<union> B) LeadsTo C)  = (F \<in> A LeadsTo C & F \<in> B LeadsTo C)"
```
```   151 by (blast intro: LeadsTo_Un LeadsTo_weaken_L)
```
```   152
```
```   153 lemma LeadsTo_UN_distrib:
```
```   154      "(F \<in> (\<Union>i \<in> I. A i) LeadsTo B)  =  (\<forall>i \<in> I. F \<in> (A i) LeadsTo B)"
```
```   155 by (blast intro: LeadsTo_UN LeadsTo_weaken_L)
```
```   156
```
```   157 lemma LeadsTo_Union_distrib:
```
```   158      "(F \<in> (Union S) LeadsTo B)  =  (\<forall>A \<in> S. F \<in> A LeadsTo B)"
```
```   159 by (blast intro: LeadsTo_Union LeadsTo_weaken_L)
```
```   160
```
```   161
```
```   162 (** More rules using the premise "Always INV" **)
```
```   163
```
```   164 lemma LeadsTo_Basis: "F \<in> A Ensures B ==> F \<in> A LeadsTo B"
```
```   165 by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)
```
```   166
```
```   167 lemma EnsuresI:
```
```   168      "[| F \<in> (A-B) Co (A \<union> B);  F \<in> transient (A-B) |]
```
```   169       ==> F \<in> A Ensures B"
```
```   170 apply (simp add: Ensures_def Constrains_eq_constrains)
```
```   171 apply (blast intro: ensuresI constrains_weaken transient_strengthen)
```
```   172 done
```
```   173
```
```   174 lemma Always_LeadsTo_Basis:
```
```   175      "[| F \<in> Always INV;
```
```   176          F \<in> (INV \<inter> (A-A')) Co (A \<union> A');
```
```   177          F \<in> transient (INV \<inter> (A-A')) |]
```
```   178   ==> F \<in> A LeadsTo A'"
```
```   179 apply (rule Always_LeadsToI, assumption)
```
```   180 apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
```
```   181 done
```
```   182
```
```   183 text{*Set difference: maybe combine with @{text leadsTo_weaken_L}??
```
```   184   This is the most useful form of the "disjunction" rule*}
```
```   185 lemma LeadsTo_Diff:
```
```   186      "[| F \<in> (A-B) LeadsTo C;  F \<in> (A \<inter> B) LeadsTo C |]
```
```   187       ==> F \<in> A LeadsTo C"
```
```   188 by (blast intro: LeadsTo_Un LeadsTo_weaken)
```
```   189
```
```   190
```
```   191 lemma LeadsTo_UN_UN:
```
```   192      "(!! i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i))
```
```   193       ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"
```
```   194 apply (simp only: Union_image_eq [symmetric])
```
```   195 apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
```
```   196 done
```
```   197
```
```   198
```
```   199 text{*Version with no index set*}
```
```   200 lemma LeadsTo_UN_UN_noindex:
```
```   201      "(!!i. F \<in> (A i) LeadsTo (A' i)) ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   202 by (blast intro: LeadsTo_UN_UN)
```
```   203
```
```   204 text{*Version with no index set*}
```
```   205 lemma all_LeadsTo_UN_UN:
```
```   206      "\<forall>i. F \<in> (A i) LeadsTo (A' i)
```
```   207       ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   208 by (blast intro: LeadsTo_UN_UN)
```
```   209
```
```   210 text{*Binary union version*}
```
```   211 lemma LeadsTo_Un_Un:
```
```   212      "[| F \<in> A LeadsTo A'; F \<in> B LeadsTo B' |]
```
```   213             ==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"
```
```   214 by (blast intro: LeadsTo_Un LeadsTo_weaken_R)
```
```   215
```
```   216
```
```   217 (** The cancellation law **)
```
```   218
```
```   219 lemma LeadsTo_cancel2:
```
```   220      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> B LeadsTo B' |]
```
```   221       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   222 by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)
```
```   223
```
```   224 lemma LeadsTo_cancel_Diff2:
```
```   225      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> (B-A') LeadsTo B' |]
```
```   226       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   227 apply (rule LeadsTo_cancel2)
```
```   228 prefer 2 apply assumption
```
```   229 apply (simp_all (no_asm_simp))
```
```   230 done
```
```   231
```
```   232 lemma LeadsTo_cancel1:
```
```   233      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> B LeadsTo B' |]
```
```   234       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   235 apply (simp add: Un_commute)
```
```   236 apply (blast intro!: LeadsTo_cancel2)
```
```   237 done
```
```   238
```
```   239 lemma LeadsTo_cancel_Diff1:
```
```   240      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> (B-A') LeadsTo B' |]
```
```   241       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   242 apply (rule LeadsTo_cancel1)
```
```   243 prefer 2 apply assumption
```
```   244 apply (simp_all (no_asm_simp))
```
```   245 done
```
```   246
```
```   247
```
```   248 text{*The impossibility law*}
```
```   249
```
```   250 text{*The set "A" may be non-empty, but it contains no reachable states*}
```
```   251 lemma LeadsTo_empty: "[|F \<in> A LeadsTo {}; all_total F|] ==> F \<in> Always (-A)"
```
```   252 apply (simp add: LeadsTo_def Always_eq_includes_reachable)
```
```   253 apply (drule leadsTo_empty, auto)
```
```   254 done
```
```   255
```
```   256
```
```   257 subsection{*PSP: Progress-Safety-Progress*}
```
```   258
```
```   259 text{*Special case of PSP: Misra's "stable conjunction"*}
```
```   260 lemma PSP_Stable:
```
```   261      "[| F \<in> A LeadsTo A';  F \<in> Stable B |]
```
```   262       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"
```
```   263 apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
```
```   264 apply (drule psp_stable, assumption)
```
```   265 apply (simp add: Int_ac)
```
```   266 done
```
```   267
```
```   268 lemma PSP_Stable2:
```
```   269      "[| F \<in> A LeadsTo A'; F \<in> Stable B |]
```
```   270       ==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"
```
```   271 by (simp add: PSP_Stable Int_ac)
```
```   272
```
```   273 lemma PSP:
```
```   274      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   275       ==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"
```
```   276 apply (simp add: LeadsTo_def Constrains_eq_constrains)
```
```   277 apply (blast dest: psp intro: leadsTo_weaken)
```
```   278 done
```
```   279
```
```   280 lemma PSP2:
```
```   281      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   282       ==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"
```
```   283 by (simp add: PSP Int_ac)
```
```   284
```
```   285 lemma PSP_Unless:
```
```   286      "[| F \<in> A LeadsTo A'; F \<in> B Unless B' |]
```
```   287       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"
```
```   288 apply (unfold Unless_def)
```
```   289 apply (drule PSP, assumption)
```
```   290 apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
```
```   291 done
```
```   292
```
```   293
```
```   294 lemma Stable_transient_Always_LeadsTo:
```
```   295      "[| F \<in> Stable A;  F \<in> transient C;
```
```   296          F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"
```
```   297 apply (erule Always_LeadsTo_weaken)
```
```   298 apply (rule LeadsTo_Diff)
```
```   299    prefer 2
```
```   300    apply (erule
```
```   301           transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])
```
```   302    apply (blast intro: subset_imp_LeadsTo)+
```
```   303 done
```
```   304
```
```   305
```
```   306 subsection{*Induction rules*}
```
```   307
```
```   308 (** Meta or object quantifier ????? **)
```
```   309 lemma LeadsTo_wf_induct:
```
```   310      "[| wf r;
```
```   311          \<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   312                     ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   313       ==> F \<in> A LeadsTo B"
```
```   314 apply (simp add: LeadsTo_eq_leadsTo)
```
```   315 apply (erule leadsTo_wf_induct)
```
```   316 apply (blast intro: leadsTo_weaken)
```
```   317 done
```
```   318
```
```   319
```
```   320 lemma Bounded_induct:
```
```   321      "[| wf r;
```
```   322          \<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   323                       ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   324       ==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"
```
```   325 apply (erule LeadsTo_wf_induct, safe)
```
```   326 apply (case_tac "m \<in> I")
```
```   327 apply (blast intro: LeadsTo_weaken)
```
```   328 apply (blast intro: subset_imp_LeadsTo)
```
```   329 done
```
```   330
```
```   331
```
```   332 lemma LessThan_induct:
```
```   333      "(!!m::nat. F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B))
```
```   334       ==> F \<in> A LeadsTo B"
```
```   335 by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
```
```   336
```
```   337 text{*Integer version.  Could generalize from 0 to any lower bound*}
```
```   338 lemma integ_0_le_induct:
```
```   339      "[| F \<in> Always {s. (0::int) \<le> f s};
```
```   340          !! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo
```
```   341                    ((A \<inter> {s. f s < z}) \<union> B) |]
```
```   342       ==> F \<in> A LeadsTo B"
```
```   343 apply (rule_tac f = "nat o f" in LessThan_induct)
```
```   344 apply (simp add: vimage_def)
```
```   345 apply (rule Always_LeadsTo_weaken, assumption+)
```
```   346 apply (auto simp add: nat_eq_iff nat_less_iff)
```
```   347 done
```
```   348
```
```   349 lemma LessThan_bounded_induct:
```
```   350      "!!l::nat. \<forall>m \<in> greaterThan l.
```
```   351                    F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)
```
```   352             ==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"
```
```   353 apply (simp only: Diff_eq [symmetric] vimage_Compl
```
```   354                   Compl_greaterThan [symmetric])
```
```   355 apply (rule wf_less_than [THEN Bounded_induct], simp)
```
```   356 done
```
```   357
```
```   358 lemma GreaterThan_bounded_induct:
```
```   359      "!!l::nat. \<forall>m \<in> lessThan l.
```
```   360                  F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)
```
```   361       ==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"
```
```   362 apply (rule_tac f = f and f1 = "%k. l - k"
```
```   363        in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
```
```   364 apply (simp add: Image_singleton, clarify)
```
```   365 apply (case_tac "m<l")
```
```   366  apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
```
```   367 apply (blast intro: not_leE subset_imp_LeadsTo)
```
```   368 done
```
```   369
```
```   370
```
```   371 subsection{*Completion: Binary and General Finite versions*}
```
```   372
```
```   373 lemma Completion:
```
```   374      "[| F \<in> A LeadsTo (A' \<union> C);  F \<in> A' Co (A' \<union> C);
```
```   375          F \<in> B LeadsTo (B' \<union> C);  F \<in> B' Co (B' \<union> C) |]
```
```   376       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
```
```   377 apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
```
```   378 apply (blast intro: completion leadsTo_weaken)
```
```   379 done
```
```   380
```
```   381 lemma Finite_completion_lemma:
```
```   382      "finite I
```
```   383       ==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->
```
```   384           (\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->
```
```   385           F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   386 apply (erule finite_induct, auto)
```
```   387 apply (rule Completion)
```
```   388    prefer 4
```
```   389    apply (simp only: INT_simps [symmetric])
```
```   390    apply (rule Constrains_INT, auto)
```
```   391 done
```
```   392
```
```   393 lemma Finite_completion:
```
```   394      "[| finite I;
```
```   395          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);
```
```   396          !!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]
```
```   397       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   398 by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])
```
```   399
```
```   400 lemma Stable_completion:
```
```   401      "[| F \<in> A LeadsTo A';  F \<in> Stable A';
```
```   402          F \<in> B LeadsTo B';  F \<in> Stable B' |]
```
```   403       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"
```
```   404 apply (unfold Stable_def)
```
```   405 apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
```
```   406 apply (force+)
```
```   407 done
```
```   408
```
```   409 lemma Finite_stable_completion:
```
```   410      "[| finite I;
```
```   411          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);
```
```   412          !!i. i \<in> I ==> F \<in> Stable (A' i) |]
```
```   413       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"
```
```   414 apply (unfold Stable_def)
```
```   415 apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
```
```   416 apply (simp_all, blast+)
```
```   417 done
```
```   418
```
```   419 end
```