src/HOL/UNITY/SubstAx.thy
author wenzelm
Thu Jul 23 22:13:42 2015 +0200 (2015-07-23)
changeset 60773 d09c66a0ea10
parent 58889 5b7a9633cfa8
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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