src/HOL/UNITY/SubstAx.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 45605 a89b4bc311a5 child 60773 d09c66a0ea10 permissions -rw-r--r--
modernized header uniformly as section;
```     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 (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]
```
```    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]
```
```    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 (unfold SUP_def)
```
```    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   using LeadsTo_UN [of "{A, B}" F id C] by auto
```
```    96
```
```    97 text{*Lets us look at the starting state*}
```
```    98 lemma single_LeadsTo_I:
```
```    99      "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"
```
```   100 by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)
```
```   101
```
```   102 lemma subset_imp_LeadsTo: "A \<subseteq> B ==> F \<in> A LeadsTo B"
```
```   103 apply (simp add: LeadsTo_def)
```
```   104 apply (blast intro: subset_imp_leadsTo)
```
```   105 done
```
```   106
```
```   107 lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, simp]
```
```   108
```
```   109 lemma LeadsTo_weaken_R:
```
```   110      "[| F \<in> A LeadsTo A';  A' \<subseteq> B' |] ==> F \<in> A LeadsTo B'"
```
```   111 apply (simp add: LeadsTo_def)
```
```   112 apply (blast intro: leadsTo_weaken_R)
```
```   113 done
```
```   114
```
```   115 lemma LeadsTo_weaken_L:
```
```   116      "[| F \<in> A LeadsTo A';  B \<subseteq> A |]
```
```   117       ==> F \<in> B LeadsTo A'"
```
```   118 apply (simp add: LeadsTo_def)
```
```   119 apply (blast intro: leadsTo_weaken_L)
```
```   120 done
```
```   121
```
```   122 lemma LeadsTo_weaken:
```
```   123      "[| F \<in> A LeadsTo A';
```
```   124          B  \<subseteq> A;   A' \<subseteq> B' |]
```
```   125       ==> F \<in> B LeadsTo B'"
```
```   126 by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)
```
```   127
```
```   128 lemma Always_LeadsTo_weaken:
```
```   129      "[| F \<in> Always C;  F \<in> A LeadsTo A';
```
```   130          C \<inter> B \<subseteq> A;   C \<inter> A' \<subseteq> B' |]
```
```   131       ==> F \<in> B LeadsTo B'"
```
```   132 by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)
```
```   133
```
```   134 (** Two theorems for "proof lattices" **)
```
```   135
```
```   136 lemma LeadsTo_Un_post: "F \<in> A LeadsTo B ==> F \<in> (A \<union> B) LeadsTo B"
```
```   137 by (blast intro: LeadsTo_Un subset_imp_LeadsTo)
```
```   138
```
```   139 lemma LeadsTo_Trans_Un:
```
```   140      "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |]
```
```   141       ==> F \<in> (A \<union> B) LeadsTo C"
```
```   142 by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)
```
```   143
```
```   144
```
```   145 (** Distributive laws **)
```
```   146
```
```   147 lemma LeadsTo_Un_distrib:
```
```   148      "(F \<in> (A \<union> B) LeadsTo C)  = (F \<in> A LeadsTo C & F \<in> B LeadsTo C)"
```
```   149 by (blast intro: LeadsTo_Un LeadsTo_weaken_L)
```
```   150
```
```   151 lemma LeadsTo_UN_distrib:
```
```   152      "(F \<in> (\<Union>i \<in> I. A i) LeadsTo B)  =  (\<forall>i \<in> I. F \<in> (A i) LeadsTo B)"
```
```   153 by (blast intro: LeadsTo_UN LeadsTo_weaken_L)
```
```   154
```
```   155 lemma LeadsTo_Union_distrib:
```
```   156      "(F \<in> (Union S) LeadsTo B)  =  (\<forall>A \<in> S. F \<in> A LeadsTo B)"
```
```   157 by (blast intro: LeadsTo_Union LeadsTo_weaken_L)
```
```   158
```
```   159
```
```   160 (** More rules using the premise "Always INV" **)
```
```   161
```
```   162 lemma LeadsTo_Basis: "F \<in> A Ensures B ==> F \<in> A LeadsTo B"
```
```   163 by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)
```
```   164
```
```   165 lemma EnsuresI:
```
```   166      "[| F \<in> (A-B) Co (A \<union> B);  F \<in> transient (A-B) |]
```
```   167       ==> F \<in> A Ensures B"
```
```   168 apply (simp add: Ensures_def Constrains_eq_constrains)
```
```   169 apply (blast intro: ensuresI constrains_weaken transient_strengthen)
```
```   170 done
```
```   171
```
```   172 lemma Always_LeadsTo_Basis:
```
```   173      "[| F \<in> Always INV;
```
```   174          F \<in> (INV \<inter> (A-A')) Co (A \<union> A');
```
```   175          F \<in> transient (INV \<inter> (A-A')) |]
```
```   176   ==> F \<in> A LeadsTo A'"
```
```   177 apply (rule Always_LeadsToI, assumption)
```
```   178 apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)
```
```   179 done
```
```   180
```
```   181 text{*Set difference: maybe combine with @{text leadsTo_weaken_L}??
```
```   182   This is the most useful form of the "disjunction" rule*}
```
```   183 lemma LeadsTo_Diff:
```
```   184      "[| F \<in> (A-B) LeadsTo C;  F \<in> (A \<inter> B) LeadsTo C |]
```
```   185       ==> F \<in> A LeadsTo C"
```
```   186 by (blast intro: LeadsTo_Un LeadsTo_weaken)
```
```   187
```
```   188
```
```   189 lemma LeadsTo_UN_UN:
```
```   190      "(!! i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i))
```
```   191       ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"
```
```   192 apply (simp only: Union_image_eq [symmetric])
```
```   193 apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)
```
```   194 done
```
```   195
```
```   196
```
```   197 text{*Version with no index set*}
```
```   198 lemma LeadsTo_UN_UN_noindex:
```
```   199      "(!!i. F \<in> (A i) LeadsTo (A' i)) ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   200 by (blast intro: LeadsTo_UN_UN)
```
```   201
```
```   202 text{*Version with no index set*}
```
```   203 lemma all_LeadsTo_UN_UN:
```
```   204      "\<forall>i. F \<in> (A i) LeadsTo (A' i)
```
```   205       ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"
```
```   206 by (blast intro: LeadsTo_UN_UN)
```
```   207
```
```   208 text{*Binary union version*}
```
```   209 lemma LeadsTo_Un_Un:
```
```   210      "[| F \<in> A LeadsTo A'; F \<in> B LeadsTo B' |]
```
```   211             ==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"
```
```   212 by (blast intro: LeadsTo_Un LeadsTo_weaken_R)
```
```   213
```
```   214
```
```   215 (** The cancellation law **)
```
```   216
```
```   217 lemma LeadsTo_cancel2:
```
```   218      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> B LeadsTo B' |]
```
```   219       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   220 by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)
```
```   221
```
```   222 lemma LeadsTo_cancel_Diff2:
```
```   223      "[| F \<in> A LeadsTo (A' \<union> B); F \<in> (B-A') LeadsTo B' |]
```
```   224       ==> F \<in> A LeadsTo (A' \<union> B')"
```
```   225 apply (rule LeadsTo_cancel2)
```
```   226 prefer 2 apply assumption
```
```   227 apply (simp_all (no_asm_simp))
```
```   228 done
```
```   229
```
```   230 lemma LeadsTo_cancel1:
```
```   231      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> B LeadsTo B' |]
```
```   232       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   233 apply (simp add: Un_commute)
```
```   234 apply (blast intro!: LeadsTo_cancel2)
```
```   235 done
```
```   236
```
```   237 lemma LeadsTo_cancel_Diff1:
```
```   238      "[| F \<in> A LeadsTo (B \<union> A'); F \<in> (B-A') LeadsTo B' |]
```
```   239       ==> F \<in> A LeadsTo (B' \<union> A')"
```
```   240 apply (rule LeadsTo_cancel1)
```
```   241 prefer 2 apply assumption
```
```   242 apply (simp_all (no_asm_simp))
```
```   243 done
```
```   244
```
```   245
```
```   246 text{*The impossibility law*}
```
```   247
```
```   248 text{*The set "A" may be non-empty, but it contains no reachable states*}
```
```   249 lemma LeadsTo_empty: "[|F \<in> A LeadsTo {}; all_total F|] ==> F \<in> Always (-A)"
```
```   250 apply (simp add: LeadsTo_def Always_eq_includes_reachable)
```
```   251 apply (drule leadsTo_empty, auto)
```
```   252 done
```
```   253
```
```   254
```
```   255 subsection{*PSP: Progress-Safety-Progress*}
```
```   256
```
```   257 text{*Special case of PSP: Misra's "stable conjunction"*}
```
```   258 lemma PSP_Stable:
```
```   259      "[| F \<in> A LeadsTo A';  F \<in> Stable B |]
```
```   260       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"
```
```   261 apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)
```
```   262 apply (drule psp_stable, assumption)
```
```   263 apply (simp add: Int_ac)
```
```   264 done
```
```   265
```
```   266 lemma PSP_Stable2:
```
```   267      "[| F \<in> A LeadsTo A'; F \<in> Stable B |]
```
```   268       ==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"
```
```   269 by (simp add: PSP_Stable Int_ac)
```
```   270
```
```   271 lemma PSP:
```
```   272      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   273       ==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"
```
```   274 apply (simp add: LeadsTo_def Constrains_eq_constrains)
```
```   275 apply (blast dest: psp intro: leadsTo_weaken)
```
```   276 done
```
```   277
```
```   278 lemma PSP2:
```
```   279      "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]
```
```   280       ==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"
```
```   281 by (simp add: PSP Int_ac)
```
```   282
```
```   283 lemma PSP_Unless:
```
```   284      "[| F \<in> A LeadsTo A'; F \<in> B Unless B' |]
```
```   285       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"
```
```   286 apply (unfold Unless_def)
```
```   287 apply (drule PSP, assumption)
```
```   288 apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)
```
```   289 done
```
```   290
```
```   291
```
```   292 lemma Stable_transient_Always_LeadsTo:
```
```   293      "[| F \<in> Stable A;  F \<in> transient C;
```
```   294          F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"
```
```   295 apply (erule Always_LeadsTo_weaken)
```
```   296 apply (rule LeadsTo_Diff)
```
```   297    prefer 2
```
```   298    apply (erule
```
```   299           transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])
```
```   300    apply (blast intro: subset_imp_LeadsTo)+
```
```   301 done
```
```   302
```
```   303
```
```   304 subsection{*Induction rules*}
```
```   305
```
```   306 (** Meta or object quantifier ????? **)
```
```   307 lemma LeadsTo_wf_induct:
```
```   308      "[| wf r;
```
```   309          \<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   310                     ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   311       ==> F \<in> A LeadsTo B"
```
```   312 apply (simp add: LeadsTo_eq_leadsTo)
```
```   313 apply (erule leadsTo_wf_induct)
```
```   314 apply (blast intro: leadsTo_weaken)
```
```   315 done
```
```   316
```
```   317
```
```   318 lemma Bounded_induct:
```
```   319      "[| wf r;
```
```   320          \<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo
```
```   321                       ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]
```
```   322       ==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"
```
```   323 apply (erule LeadsTo_wf_induct, safe)
```
```   324 apply (case_tac "m \<in> I")
```
```   325 apply (blast intro: LeadsTo_weaken)
```
```   326 apply (blast intro: subset_imp_LeadsTo)
```
```   327 done
```
```   328
```
```   329
```
```   330 lemma LessThan_induct:
```
```   331      "(!!m::nat. F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B))
```
```   332       ==> F \<in> A LeadsTo B"
```
```   333 by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)
```
```   334
```
```   335 text{*Integer version.  Could generalize from 0 to any lower bound*}
```
```   336 lemma integ_0_le_induct:
```
```   337      "[| F \<in> Always {s. (0::int) \<le> f s};
```
```   338          !! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo
```
```   339                    ((A \<inter> {s. f s < z}) \<union> B) |]
```
```   340       ==> F \<in> A LeadsTo B"
```
```   341 apply (rule_tac f = "nat o f" in LessThan_induct)
```
```   342 apply (simp add: vimage_def)
```
```   343 apply (rule Always_LeadsTo_weaken, assumption+)
```
```   344 apply (auto simp add: nat_eq_iff nat_less_iff)
```
```   345 done
```
```   346
```
```   347 lemma LessThan_bounded_induct:
```
```   348      "!!l::nat. \<forall>m \<in> greaterThan l.
```
```   349                    F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)
```
```   350             ==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"
```
```   351 apply (simp only: Diff_eq [symmetric] vimage_Compl
```
```   352                   Compl_greaterThan [symmetric])
```
```   353 apply (rule wf_less_than [THEN Bounded_induct], simp)
```
```   354 done
```
```   355
```
```   356 lemma GreaterThan_bounded_induct:
```
```   357      "!!l::nat. \<forall>m \<in> lessThan l.
```
```   358                  F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)
```
```   359       ==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"
```
```   360 apply (rule_tac f = f and f1 = "%k. l - k"
```
```   361        in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])
```
```   362 apply (simp add: Image_singleton, clarify)
```
```   363 apply (case_tac "m<l")
```
```   364  apply (blast intro: LeadsTo_weaken_R diff_less_mono2)
```
```   365 apply (blast intro: not_leE subset_imp_LeadsTo)
```
```   366 done
```
```   367
```
```   368
```
```   369 subsection{*Completion: Binary and General Finite versions*}
```
```   370
```
```   371 lemma Completion:
```
```   372      "[| F \<in> A LeadsTo (A' \<union> C);  F \<in> A' Co (A' \<union> C);
```
```   373          F \<in> B LeadsTo (B' \<union> C);  F \<in> B' Co (B' \<union> C) |]
```
```   374       ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"
```
```   375 apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)
```
```   376 apply (blast intro: completion leadsTo_weaken)
```
```   377 done
```
```   378
```
```   379 lemma Finite_completion_lemma:
```
```   380      "finite I
```
```   381       ==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->
```
```   382           (\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->
```
```   383           F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   384 apply (erule finite_induct, auto)
```
```   385 apply (rule Completion)
```
```   386    prefer 4
```
```   387    apply (simp only: INT_simps [symmetric])
```
```   388    apply (rule Constrains_INT, auto)
```
```   389 done
```
```   390
```
```   391 lemma Finite_completion:
```
```   392      "[| finite I;
```
```   393          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);
```
```   394          !!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]
```
```   395       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"
```
```   396 by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])
```
```   397
```
```   398 lemma Stable_completion:
```
```   399      "[| F \<in> A LeadsTo A';  F \<in> Stable A';
```
```   400          F \<in> B LeadsTo B';  F \<in> Stable B' |]
```
```   401       ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"
```
```   402 apply (unfold Stable_def)
```
```   403 apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])
```
```   404 apply (force+)
```
```   405 done
```
```   406
```
```   407 lemma Finite_stable_completion:
```
```   408      "[| finite I;
```
```   409          !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);
```
```   410          !!i. i \<in> I ==> F \<in> Stable (A' i) |]
```
```   411       ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"
```
```   412 apply (unfold Stable_def)
```
```   413 apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])
```
```   414 apply (simp_all, blast+)
```
```   415 done
```
```   416
```
```   417 end
```