src/HOL/UNITY/SubstAx.thy
 author wenzelm Thu Jul 23 22:13:42 2015 +0200 (2015-07-23) changeset 60773 d09c66a0ea10 parent 58889 5b7a9633cfa8 child 61824 dcbe9f756ae0 permissions -rw-r--r--
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```     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 (unfold SUP_def)
```
```    88 apply (blast intro: LeadsTo_Union)
```
```    89 done
```
```    90
```
```    91 text{*Binary union introduction rule*}
```
```    92 lemma LeadsTo_Un:
```
```    93      "[| F \<in> A LeadsTo C; F \<in> B LeadsTo C |] ==> F \<in> (A \<union> B) LeadsTo C"
```
```    94   using LeadsTo_UN [of "{A, B}" F id C] by auto
```
```    95
```
```    96 text{*Lets us look at the starting state*}
```
```    97 lemma single_LeadsTo_I:
```
```    98      "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"
```
```    99 by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)
```
```   100
```
```   101 lemma subset_imp_LeadsTo: "A \<subseteq> B ==> F \<in> A LeadsTo B"
```
```   102 apply (simp add: LeadsTo_def)
```
```   103 apply (blast intro: subset_imp_leadsTo)
```
```   104 done
```
```   105
```
```   106 lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, simp]
```
```   107
```
```   108 lemma LeadsTo_weaken_R:
```
```   109      "[| F \<in> A LeadsTo A';  A' \<subseteq> B' |] ==> F \<in> A LeadsTo B'"
```
```   110 apply (simp add: LeadsTo_def)
```
```   111 apply (blast intro: leadsTo_weaken_R)
```
```   112 done
```
```   113
```
```   114 lemma LeadsTo_weaken_L:
```
```   115      "[| F \<in> A LeadsTo A';  B \<subseteq> A |]
```
```   116       ==> F \<in> B LeadsTo A'"
```
```   117 apply (simp add: LeadsTo_def)
```
```   118 apply (blast intro: leadsTo_weaken_L)
```
```   119 done
```
```   120
```
```   121 lemma LeadsTo_weaken:
```
```   122      "[| F \<in> A LeadsTo A';
```
```   123          B  \<subseteq> A;   A' \<subseteq> B' |]
```
```   124       ==> F \<in> B LeadsTo B'"
```
```   125 by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)
```
```   126
```
```   127 lemma Always_LeadsTo_weaken:
```
```   128      "[| F \<in> Always C;  F \<in> A LeadsTo A';
```
```   129          C \<inter> B \<subseteq> A;   C \<inter> A' \<subseteq> B' |]
```
```   130       ==> F \<in> B LeadsTo B'"
```
```   131 by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)
```
```   132
```
```   133 (** Two theorems for "proof lattices" **)
```
```   134
```
```   135 lemma LeadsTo_Un_post: "F \<in> A LeadsTo B ==> F \<in> (A \<union> B) LeadsTo B"
```
```   136 by (blast intro: LeadsTo_Un subset_imp_LeadsTo)
```
```   137
```
```   138 lemma LeadsTo_Trans_Un:
```
```   139      "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |]
```
```   140       ==> F \<in> (A \<union> B) LeadsTo C"
```
```   141 by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)
```
```   142
```
```   143
```
```   144 (** Distributive laws **)
```
```   145
```
```   146 lemma LeadsTo_Un_distrib:
```
```   147      "(F \<in> (A \<union> B) LeadsTo C)  = (F \<in> A LeadsTo C & F \<in> B LeadsTo C)"
```
```   148 by (blast intro: LeadsTo_Un LeadsTo_weaken_L)
```
```   149
```
```   150 lemma LeadsTo_UN_distrib:
```
```   151      "(F \<in> (\<Union>i \<in> I. A i) LeadsTo B)  =  (\<forall>i \<in> I. F \<in> (A i) LeadsTo B)"
```
```   152 by (blast intro: LeadsTo_UN LeadsTo_weaken_L)
```
```   153
```
```   154 lemma LeadsTo_Union_distrib:
```
```   155      "(F \<in> (Union S) LeadsTo B)  =  (\<forall>A \<in> S. F \<in> A LeadsTo B)"
```
```   156 by (blast intro: LeadsTo_Union LeadsTo_weaken_L)
```
```   157
```
```   158
```
```   159 (** More rules using the premise "Always INV" **)
```
```   160
```
```   161 lemma LeadsTo_Basis: "F \<in> A Ensures B ==> F \<in> A LeadsTo B"
```
```   162 by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)
```
```   163
```
```   164 lemma EnsuresI:
```
```   165      "[| F \<in> (A-B) Co (A \<union> B);  F \<in> transient (A-B) |]
```
```   166       ==> F \<in> A Ensures B"
```
```   167 apply (simp add: Ensures_def Constrains_eq_constrains)
```
```   168 apply (blast intro: ensuresI constrains_weaken transient_strengthen)
```
```   169 done
```
```   170
```
```   171 lemma Always_LeadsTo_Basis:
```
```   172      "[| F \<in> Always INV;
```
```   173          F \<in> (INV \<inter> (A-A')) Co (A \<union> A');
```
```   174          F \<in> transient (INV \<inter> (A-A')) |]
```
```   175   ==> F \<in> A LeadsTo A'"
```
```   176 apply (rule Always_LeadsToI, assumption)
```
```   177 apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
```
```   178 done
```
```   179
```
```   180 text{*Set difference: maybe combine with @{text leadsTo_weaken_L}??
```
```   181   This is the most useful form of the "disjunction" rule*}
```
```   182 lemma LeadsTo_Diff:
```
```   183      "[| F \<in> (A-B) LeadsTo C;  F \<in> (A \<inter> B) LeadsTo C |]
```
```   184       ==> F \<in> A LeadsTo C"
```
```   185 by (blast intro: LeadsTo_Un LeadsTo_weaken)
```
```   186
```
```   187
```
```   188 lemma LeadsTo_UN_UN:
```
```   189      "(!! i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i))
```
```   190       ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"
```
```   191 apply (simp only: Union_image_eq [symmetric])
```
```   192 apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
```
```   193 done
```
```   194
```
```   195
```
```   196 text{*Version with no index set*}
```
```   197 lemma LeadsTo_UN_UN_noindex:
```
```   198      "(!!i. F \<in> (A i) LeadsTo (A' i)) ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   199 by (blast intro: LeadsTo_UN_UN)
```
```   200
```
```   201 text{*Version with no index set*}
```
```   202 lemma all_LeadsTo_UN_UN:
```
```   203      "\<forall>i. F \<in> (A i) LeadsTo (A' i)
```
```   204       ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   205 by (blast intro: LeadsTo_UN_UN)
```
```   206
```
```   207 text{*Binary union version*}
```
```   208 lemma LeadsTo_Un_Un:
```
```   209      "[| F \<in> A LeadsTo A'; F \<in> B LeadsTo B' |]
```
```   210             ==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"
```
```   211 by (blast intro: LeadsTo_Un LeadsTo_weaken_R)
```
```   212
```
```   213
```
```   214 (** The cancellation law **)
```
```   215
```
```   216 lemma LeadsTo_cancel2:
```
```   217      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> B LeadsTo B' |]
```
```   218       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   219 by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)
```
```   220
```
```   221 lemma LeadsTo_cancel_Diff2:
```
```   222      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> (B-A') LeadsTo B' |]
```
```   223       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   224 apply (rule LeadsTo_cancel2)
```
```   225 prefer 2 apply assumption
```
```   226 apply (simp_all (no_asm_simp))
```
```   227 done
```
```   228
```
```   229 lemma LeadsTo_cancel1:
```
```   230      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> B LeadsTo B' |]
```
```   231       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   232 apply (simp add: Un_commute)
```
```   233 apply (blast intro!: LeadsTo_cancel2)
```
```   234 done
```
```   235
```
```   236 lemma LeadsTo_cancel_Diff1:
```
```   237      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> (B-A') LeadsTo B' |]
```
```   238       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   239 apply (rule LeadsTo_cancel1)
```
```   240 prefer 2 apply assumption
```
```   241 apply (simp_all (no_asm_simp))
```
```   242 done
```
```   243
```
```   244
```
```   245 text{*The impossibility law*}
```
```   246
```
```   247 text{*The set "A" may be non-empty, but it contains no reachable states*}
```
```   248 lemma LeadsTo_empty: "[|F \<in> A LeadsTo {}; all_total F|] ==> F \<in> Always (-A)"
```
```   249 apply (simp add: LeadsTo_def Always_eq_includes_reachable)
```
```   250 apply (drule leadsTo_empty, auto)
```
```   251 done
```
```   252
```
```   253
```
```   254 subsection{*PSP: Progress-Safety-Progress*}
```
```   255
```
```   256 text{*Special case of PSP: Misra's "stable conjunction"*}
```
```   257 lemma PSP_Stable:
```
```   258      "[| F \<in> A LeadsTo A';  F \<in> Stable B |]
```
```   259       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"
```
```   260 apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
```
```   261 apply (drule psp_stable, assumption)
```
```   262 apply (simp add: Int_ac)
```
```   263 done
```
```   264
```
```   265 lemma PSP_Stable2:
```
```   266      "[| F \<in> A LeadsTo A'; F \<in> Stable B |]
```
```   267       ==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"
```
```   268 by (simp add: PSP_Stable Int_ac)
```
```   269
```
```   270 lemma PSP:
```
```   271      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   272       ==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"
```
```   273 apply (simp add: LeadsTo_def Constrains_eq_constrains)
```
```   274 apply (blast dest: psp intro: leadsTo_weaken)
```
```   275 done
```
```   276
```
```   277 lemma PSP2:
```
```   278      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   279       ==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"
```
```   280 by (simp add: PSP Int_ac)
```
```   281
```
```   282 lemma PSP_Unless:
```
```   283      "[| F \<in> A LeadsTo A'; F \<in> B Unless B' |]
```
```   284       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"
```
```   285 apply (unfold Unless_def)
```
```   286 apply (drule PSP, assumption)
```
```   287 apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
```
```   288 done
```
```   289
```
```   290
```
```   291 lemma Stable_transient_Always_LeadsTo:
```
```   292      "[| F \<in> Stable A;  F \<in> transient C;
```
```   293          F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"
```
```   294 apply (erule Always_LeadsTo_weaken)
```
```   295 apply (rule LeadsTo_Diff)
```
```   296    prefer 2
```
```   297    apply (erule
```
```   298           transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])
```
```   299    apply (blast intro: subset_imp_LeadsTo)+
```
```   300 done
```
```   301
```
```   302
```
```   303 subsection{*Induction rules*}
```
```   304
```
```   305 (** Meta or object quantifier ????? **)
```
```   306 lemma LeadsTo_wf_induct:
```
```   307      "[| wf r;
```
```   308          \<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   309                     ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   310       ==> F \<in> A LeadsTo B"
```
```   311 apply (simp add: LeadsTo_eq_leadsTo)
```
```   312 apply (erule leadsTo_wf_induct)
```
```   313 apply (blast intro: leadsTo_weaken)
```
```   314 done
```
```   315
```
```   316
```
```   317 lemma Bounded_induct:
```
```   318      "[| wf r;
```
```   319          \<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   320                       ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   321       ==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"
```
```   322 apply (erule LeadsTo_wf_induct, safe)
```
```   323 apply (case_tac "m \<in> I")
```
```   324 apply (blast intro: LeadsTo_weaken)
```
```   325 apply (blast intro: subset_imp_LeadsTo)
```
```   326 done
```
```   327
```
```   328
```
```   329 lemma LessThan_induct:
```
```   330      "(!!m::nat. F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B))
```
```   331       ==> F \<in> A LeadsTo B"
```
```   332 by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
```
```   333
```
```   334 text{*Integer version.  Could generalize from 0 to any lower bound*}
```
```   335 lemma integ_0_le_induct:
```
```   336      "[| F \<in> Always {s. (0::int) \<le> f s};
```
```   337          !! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo
```
```   338                    ((A \<inter> {s. f s < z}) \<union> B) |]
```
```   339       ==> F \<in> A LeadsTo B"
```
```   340 apply (rule_tac f = "nat o f" in LessThan_induct)
```
```   341 apply (simp add: vimage_def)
```
```   342 apply (rule Always_LeadsTo_weaken, assumption+)
```
```   343 apply (auto simp add: nat_eq_iff nat_less_iff)
```
```   344 done
```
```   345
```
```   346 lemma LessThan_bounded_induct:
```
```   347      "!!l::nat. \<forall>m \<in> greaterThan l.
```
```   348                    F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)
```
```   349             ==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"
```
```   350 apply (simp only: Diff_eq [symmetric] vimage_Compl
```
```   351                   Compl_greaterThan [symmetric])
```
```   352 apply (rule wf_less_than [THEN Bounded_induct], simp)
```
```   353 done
```
```   354
```
```   355 lemma GreaterThan_bounded_induct:
```
```   356      "!!l::nat. \<forall>m \<in> lessThan l.
```
```   357                  F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)
```
```   358       ==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"
```
```   359 apply (rule_tac f = f and f1 = "%k. l - k"
```
```   360        in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
```
```   361 apply (simp add: Image_singleton, clarify)
```
```   362 apply (case_tac "m<l")
```
```   363  apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
```
```   364 apply (blast intro: not_leE subset_imp_LeadsTo)
```
```   365 done
```
```   366
```
```   367
```
```   368 subsection{*Completion: Binary and General Finite versions*}
```
```   369
```
```   370 lemma Completion:
```
```   371      "[| F \<in> A LeadsTo (A' \<union> C);  F \<in> A' Co (A' \<union> C);
```
```   372          F \<in> B LeadsTo (B' \<union> C);  F \<in> B' Co (B' \<union> C) |]
```
```   373       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
```
```   374 apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
```
```   375 apply (blast intro: completion leadsTo_weaken)
```
```   376 done
```
```   377
```
```   378 lemma Finite_completion_lemma:
```
```   379      "finite I
```
```   380       ==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->
```
```   381           (\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->
```
```   382           F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   383 apply (erule finite_induct, auto)
```
```   384 apply (rule Completion)
```
```   385    prefer 4
```
```   386    apply (simp only: INT_simps [symmetric])
```
```   387    apply (rule Constrains_INT, auto)
```
```   388 done
```
```   389
```
```   390 lemma Finite_completion:
```
```   391      "[| finite I;
```
```   392          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);
```
```   393          !!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]
```
```   394       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   395 by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])
```
```   396
```
```   397 lemma Stable_completion:
```
```   398      "[| F \<in> A LeadsTo A';  F \<in> Stable A';
```
```   399          F \<in> B LeadsTo B';  F \<in> Stable B' |]
```
```   400       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"
```
```   401 apply (unfold Stable_def)
```
```   402 apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
```
```   403 apply (force+)
```
```   404 done
```
```   405
```
```   406 lemma Finite_stable_completion:
```
```   407      "[| finite I;
```
```   408          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);
```
```   409          !!i. i \<in> I ==> F \<in> Stable (A' i) |]
```
```   410       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"
```
```   411 apply (unfold Stable_def)
```
```   412 apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
```
```   413 apply (simp_all, blast+)
```
```   414 done
```
```   415
```
```   416 end
```