src/HOL/UNITY/ELT.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 62343 24106dc44def
child 63146 f1ecba0272f9
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
     1 (*  Title:      HOL/UNITY/ELT.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1999  University of Cambridge
     4 
     5 leadsTo strengthened with a specification of the allowable sets transient parts
     6 
     7 TRY INSTEAD (to get rid of the {} and to gain strong induction)
     8 
     9   elt :: "['a set set, 'a program, 'a set] => ('a set) set"
    10 
    11 inductive "elt CC F B"
    12   intros 
    13 
    14     Weaken:  "A <= B ==> A : elt CC F B"
    15 
    16     ETrans:  "[| F : A ensures A';  A-A' : CC;  A' : elt CC F B |]
    17               ==> A : elt CC F B"
    18 
    19     Union:  "{A. A: S} : Pow (elt CC F B) ==> (Union S) : elt CC F B"
    20 
    21   monos Pow_mono
    22 *)
    23 
    24 section{*Progress Under Allowable Sets*}
    25 
    26 theory ELT imports Project begin
    27 
    28 inductive_set
    29   (*LEADS-TO constant for the inductive definition*)
    30   elt :: "['a set set, 'a program] => ('a set * 'a set) set"
    31   for CC :: "'a set set" and F :: "'a program"
    32  where
    33 
    34    Basis:  "[| F : A ensures B;  A-B : (insert {} CC) |] ==> (A,B) : elt CC F"
    35 
    36  | Trans:  "[| (A,B) : elt CC F;  (B,C) : elt CC F |] ==> (A,C) : elt CC F"
    37 
    38  | Union:  "ALL A: S. (A,B) : elt CC F ==> (Union S, B) : elt CC F"
    39 
    40 
    41 definition  
    42   (*the set of all sets determined by f alone*)
    43   givenBy :: "['a => 'b] => 'a set set"
    44   where "givenBy f = range (%B. f-` B)"
    45 
    46 definition
    47   (*visible version of the LEADS-TO relation*)
    48   leadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
    49                                         ("(3_/ leadsTo[_]/ _)" [80,0,80] 80)
    50   where "leadsETo A CC B = {F. (A,B) : elt CC F}"
    51 
    52 definition
    53   LeadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
    54                                         ("(3_/ LeadsTo[_]/ _)" [80,0,80] 80)
    55   where "LeadsETo A CC B =
    56       {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC] B}"
    57 
    58 
    59 (*** givenBy ***)
    60 
    61 lemma givenBy_id [simp]: "givenBy id = UNIV"
    62 by (unfold givenBy_def, auto)
    63 
    64 lemma givenBy_eq_all: "(givenBy v) = {A. ALL x:A. ALL y. v x = v y --> y: A}"
    65 apply (unfold givenBy_def, safe)
    66 apply (rule_tac [2] x = "v ` _" in image_eqI, auto)
    67 done
    68 
    69 lemma givenByI: "(!!x y. [| x:A;  v x = v y |] ==> y: A) ==> A: givenBy v"
    70 by (subst givenBy_eq_all, blast)
    71 
    72 lemma givenByD: "[| A: givenBy v;  x:A;  v x = v y |] ==> y: A"
    73 by (unfold givenBy_def, auto)
    74 
    75 lemma empty_mem_givenBy [iff]: "{} : givenBy v"
    76 by (blast intro!: givenByI)
    77 
    78 lemma givenBy_imp_eq_Collect: "A: givenBy v ==> EX P. A = {s. P(v s)}"
    79 apply (rule_tac x = "%n. EX s. v s = n & s : A" in exI)
    80 apply (simp (no_asm_use) add: givenBy_eq_all)
    81 apply blast
    82 done
    83 
    84 lemma Collect_mem_givenBy: "{s. P(v s)} : givenBy v"
    85 by (unfold givenBy_def, best)
    86 
    87 lemma givenBy_eq_Collect: "givenBy v = {A. EX P. A = {s. P(v s)}}"
    88 by (blast intro: Collect_mem_givenBy givenBy_imp_eq_Collect)
    89 
    90 (*preserving v preserves properties given by v*)
    91 lemma preserves_givenBy_imp_stable:
    92      "[| F : preserves v;  D : givenBy v |] ==> F : stable D"
    93 by (force simp add: preserves_subset_stable [THEN subsetD] givenBy_eq_Collect)
    94 
    95 lemma givenBy_o_subset: "givenBy (w o v) <= givenBy v"
    96 apply (simp (no_asm) add: givenBy_eq_Collect)
    97 apply best 
    98 done
    99 
   100 lemma givenBy_DiffI:
   101      "[| A : givenBy v;  B : givenBy v |] ==> A-B : givenBy v"
   102 apply (simp (no_asm_use) add: givenBy_eq_Collect)
   103 apply safe
   104 apply (rule_tac x = "%z. R z & ~ Q z" for R Q in exI)
   105 unfolding set_diff_eq
   106 apply auto
   107 done
   108 
   109 
   110 (** Standard leadsTo rules **)
   111 
   112 lemma leadsETo_Basis [intro]: 
   113      "[| F: A ensures B;  A-B: insert {} CC |] ==> F : A leadsTo[CC] B"
   114 apply (unfold leadsETo_def)
   115 apply (blast intro: elt.Basis)
   116 done
   117 
   118 lemma leadsETo_Trans: 
   119      "[| F : A leadsTo[CC] B;  F : B leadsTo[CC] C |] ==> F : A leadsTo[CC] C"
   120 apply (unfold leadsETo_def)
   121 apply (blast intro: elt.Trans)
   122 done
   123 
   124 
   125 (*Useful with cancellation, disjunction*)
   126 lemma leadsETo_Un_duplicate:
   127      "F : A leadsTo[CC] (A' Un A') ==> F : A leadsTo[CC] A'"
   128 by (simp add: Un_ac)
   129 
   130 lemma leadsETo_Un_duplicate2:
   131      "F : A leadsTo[CC] (A' Un C Un C) ==> F : A leadsTo[CC] (A' Un C)"
   132 by (simp add: Un_ac)
   133 
   134 (*The Union introduction rule as we should have liked to state it*)
   135 lemma leadsETo_Union:
   136     "(!!A. A : S ==> F : A leadsTo[CC] B) ==> F : (\<Union>S) leadsTo[CC] B"
   137 apply (unfold leadsETo_def)
   138 apply (blast intro: elt.Union)
   139 done
   140 
   141 lemma leadsETo_UN:
   142     "(!!i. i : I ==> F : (A i) leadsTo[CC] B)  
   143      ==> F : (UN i:I. A i) leadsTo[CC] B"
   144 apply (blast intro: leadsETo_Union)
   145 done
   146 
   147 (*The INDUCTION rule as we should have liked to state it*)
   148 lemma leadsETo_induct:
   149   "[| F : za leadsTo[CC] zb;   
   150       !!A B. [| F : A ensures B;  A-B : insert {} CC |] ==> P A B;  
   151       !!A B C. [| F : A leadsTo[CC] B; P A B; F : B leadsTo[CC] C; P B C |]  
   152                ==> P A C;  
   153       !!B S. ALL A:S. F : A leadsTo[CC] B & P A B ==> P (\<Union>S) B  
   154    |] ==> P za zb"
   155 apply (unfold leadsETo_def)
   156 apply (drule CollectD) 
   157 apply (erule elt.induct, blast+)
   158 done
   159 
   160 
   161 (** New facts involving leadsETo **)
   162 
   163 lemma leadsETo_mono: "CC' <= CC ==> (A leadsTo[CC'] B) <= (A leadsTo[CC] B)"
   164 apply safe
   165 apply (erule leadsETo_induct)
   166 prefer 3 apply (blast intro: leadsETo_Union)
   167 prefer 2 apply (blast intro: leadsETo_Trans)
   168 apply blast
   169 done
   170 
   171 lemma leadsETo_Trans_Un:
   172      "[| F : A leadsTo[CC] B;  F : B leadsTo[DD] C |]  
   173       ==> F : A leadsTo[CC Un DD] C"
   174 by (blast intro: leadsETo_mono [THEN subsetD] leadsETo_Trans)
   175 
   176 lemma leadsETo_Union_Int:
   177  "(!!A. A : S ==> F : (A Int C) leadsTo[CC] B) 
   178   ==> F : (\<Union>S Int C) leadsTo[CC] B"
   179 apply (unfold leadsETo_def)
   180 apply (simp only: Int_Union_Union)
   181 apply (blast intro: elt.Union)
   182 done
   183 
   184 (*Binary union introduction rule*)
   185 lemma leadsETo_Un:
   186      "[| F : A leadsTo[CC] C; F : B leadsTo[CC] C |] 
   187       ==> F : (A Un B) leadsTo[CC] C"
   188   using leadsETo_Union [of "{A, B}" F CC C] by auto
   189 
   190 lemma single_leadsETo_I:
   191      "(!!x. x : A ==> F : {x} leadsTo[CC] B) ==> F : A leadsTo[CC] B"
   192 by (subst UN_singleton [symmetric], rule leadsETo_UN, blast)
   193 
   194 
   195 lemma subset_imp_leadsETo: "A<=B ==> F : A leadsTo[CC] B"
   196 by (simp add: subset_imp_ensures [THEN leadsETo_Basis] 
   197               Diff_eq_empty_iff [THEN iffD2])
   198 
   199 lemmas empty_leadsETo = empty_subsetI [THEN subset_imp_leadsETo, simp]
   200 
   201 
   202 
   203 (** Weakening laws **)
   204 
   205 lemma leadsETo_weaken_R:
   206      "[| F : A leadsTo[CC] A';  A'<=B' |] ==> F : A leadsTo[CC] B'"
   207 by (blast intro: subset_imp_leadsETo leadsETo_Trans)
   208 
   209 lemma leadsETo_weaken_L:
   210      "[| F : A leadsTo[CC] A'; B<=A |] ==> F : B leadsTo[CC] A'"
   211 by (blast intro: leadsETo_Trans subset_imp_leadsETo)
   212 
   213 (*Distributes over binary unions*)
   214 lemma leadsETo_Un_distrib:
   215      "F : (A Un B) leadsTo[CC] C  =   
   216       (F : A leadsTo[CC] C & F : B leadsTo[CC] C)"
   217 by (blast intro: leadsETo_Un leadsETo_weaken_L)
   218 
   219 lemma leadsETo_UN_distrib:
   220      "F : (UN i:I. A i) leadsTo[CC] B  =   
   221       (ALL i : I. F : (A i) leadsTo[CC] B)"
   222 by (blast intro: leadsETo_UN leadsETo_weaken_L)
   223 
   224 lemma leadsETo_Union_distrib:
   225      "F : (\<Union>S) leadsTo[CC] B  =  (ALL A : S. F : A leadsTo[CC] B)"
   226 by (blast intro: leadsETo_Union leadsETo_weaken_L)
   227 
   228 lemma leadsETo_weaken:
   229      "[| F : A leadsTo[CC'] A'; B<=A; A'<=B';  CC' <= CC |]  
   230       ==> F : B leadsTo[CC] B'"
   231 apply (drule leadsETo_mono [THEN subsetD], assumption)
   232 apply (blast del: subsetCE 
   233              intro: leadsETo_weaken_R leadsETo_weaken_L leadsETo_Trans)
   234 done
   235 
   236 lemma leadsETo_givenBy:
   237      "[| F : A leadsTo[CC] A';  CC <= givenBy v |]  
   238       ==> F : A leadsTo[givenBy v] A'"
   239 by (blast intro: leadsETo_weaken)
   240 
   241 
   242 (*Set difference*)
   243 lemma leadsETo_Diff:
   244      "[| F : (A-B) leadsTo[CC] C; F : B leadsTo[CC] C |]  
   245       ==> F : A leadsTo[CC] C"
   246 by (blast intro: leadsETo_Un leadsETo_weaken)
   247 
   248 
   249 (*Binary union version*)
   250 lemma leadsETo_Un_Un:
   251      "[| F : A leadsTo[CC] A';  F : B leadsTo[CC] B' |]  
   252       ==> F : (A Un B) leadsTo[CC] (A' Un B')"
   253 by (blast intro: leadsETo_Un leadsETo_weaken_R)
   254 
   255 
   256 (** The cancellation law **)
   257 
   258 lemma leadsETo_cancel2:
   259      "[| F : A leadsTo[CC] (A' Un B); F : B leadsTo[CC] B' |]  
   260       ==> F : A leadsTo[CC] (A' Un B')"
   261 by (blast intro: leadsETo_Un_Un subset_imp_leadsETo leadsETo_Trans)
   262 
   263 lemma leadsETo_cancel1:
   264      "[| F : A leadsTo[CC] (B Un A'); F : B leadsTo[CC] B' |]  
   265     ==> F : A leadsTo[CC] (B' Un A')"
   266 apply (simp add: Un_commute)
   267 apply (blast intro!: leadsETo_cancel2)
   268 done
   269 
   270 lemma leadsETo_cancel_Diff1:
   271      "[| F : A leadsTo[CC] (B Un A'); F : (B-A') leadsTo[CC] B' |]  
   272     ==> F : A leadsTo[CC] (B' Un A')"
   273 apply (rule leadsETo_cancel1)
   274  prefer 2 apply assumption
   275 apply simp_all
   276 done
   277 
   278 
   279 (** PSP: Progress-Safety-Progress **)
   280 
   281 (*Special case of PSP: Misra's "stable conjunction"*)
   282 lemma e_psp_stable: 
   283    "[| F : A leadsTo[CC] A';  F : stable B;  ALL C:CC. C Int B : CC |]  
   284     ==> F : (A Int B) leadsTo[CC] (A' Int B)"
   285 apply (unfold stable_def)
   286 apply (erule leadsETo_induct)
   287 prefer 3 apply (blast intro: leadsETo_Union_Int)
   288 prefer 2 apply (blast intro: leadsETo_Trans)
   289 apply (rule leadsETo_Basis)
   290 prefer 2 apply (force simp add: Diff_Int_distrib2 [symmetric])
   291 apply (simp add: ensures_def Diff_Int_distrib2 [symmetric] 
   292                  Int_Un_distrib2 [symmetric])
   293 apply (blast intro: transient_strengthen constrains_Int)
   294 done
   295 
   296 lemma e_psp_stable2:
   297      "[| F : A leadsTo[CC] A'; F : stable B;  ALL C:CC. C Int B : CC |]  
   298       ==> F : (B Int A) leadsTo[CC] (B Int A')"
   299 by (simp (no_asm_simp) add: e_psp_stable Int_ac)
   300 
   301 lemma e_psp:
   302      "[| F : A leadsTo[CC] A'; F : B co B';   
   303          ALL C:CC. C Int B Int B' : CC |]  
   304       ==> F : (A Int B') leadsTo[CC] ((A' Int B) Un (B' - B))"
   305 apply (erule leadsETo_induct)
   306 prefer 3 apply (blast intro: leadsETo_Union_Int)
   307 (*Transitivity case has a delicate argument involving "cancellation"*)
   308 apply (rule_tac [2] leadsETo_Un_duplicate2)
   309 apply (erule_tac [2] leadsETo_cancel_Diff1)
   310 prefer 2
   311  apply (simp add: Int_Diff Diff_triv)
   312  apply (blast intro: leadsETo_weaken_L dest: constrains_imp_subset)
   313 (*Basis case*)
   314 apply (rule leadsETo_Basis)
   315 apply (blast intro: psp_ensures)
   316 apply (subgoal_tac "A Int B' - (Ba Int B Un (B' - B)) = (A - Ba) Int B Int B'")
   317 apply auto
   318 done
   319 
   320 lemma e_psp2:
   321      "[| F : A leadsTo[CC] A'; F : B co B';   
   322          ALL C:CC. C Int B Int B' : CC |]  
   323       ==> F : (B' Int A) leadsTo[CC] ((B Int A') Un (B' - B))"
   324 by (simp add: e_psp Int_ac)
   325 
   326 
   327 (*** Special properties involving the parameter [CC] ***)
   328 
   329 (*??IS THIS NEEDED?? or is it just an example of what's provable??*)
   330 lemma gen_leadsETo_imp_Join_leadsETo:
   331      "[| F: (A leadsTo[givenBy v] B);  G : preserves v;   
   332          F\<squnion>G : stable C |]  
   333       ==> F\<squnion>G : ((C Int A) leadsTo[(%D. C Int D) ` givenBy v] B)"
   334 apply (erule leadsETo_induct)
   335   prefer 3
   336   apply (subst Int_Union) 
   337   apply (blast intro: leadsETo_UN)
   338 prefer 2
   339  apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
   340 apply (rule leadsETo_Basis)
   341 apply (auto simp add: Diff_eq_empty_iff [THEN iffD2] 
   342                       Int_Diff ensures_def givenBy_eq_Collect)
   343 prefer 3 apply (blast intro: transient_strengthen)
   344 apply (drule_tac [2] P1 = P in preserves_subset_stable [THEN subsetD])
   345 apply (drule_tac P1 = P in preserves_subset_stable [THEN subsetD])
   346 apply (unfold stable_def)
   347 apply (blast intro: constrains_Int [THEN constrains_weaken])+
   348 done
   349 
   350 (**** Relationship with traditional "leadsTo", strong & weak ****)
   351 
   352 (** strong **)
   353 
   354 lemma leadsETo_subset_leadsTo: "(A leadsTo[CC] B) <= (A leadsTo B)"
   355 apply safe
   356 apply (erule leadsETo_induct)
   357   prefer 3 apply (blast intro: leadsTo_Union)
   358  prefer 2 apply (blast intro: leadsTo_Trans, blast)
   359 done
   360 
   361 lemma leadsETo_UNIV_eq_leadsTo: "(A leadsTo[UNIV] B) = (A leadsTo B)"
   362 apply safe
   363 apply (erule leadsETo_subset_leadsTo [THEN subsetD])
   364 (*right-to-left case*)
   365 apply (erule leadsTo_induct)
   366   prefer 3 apply (blast intro: leadsETo_Union)
   367  prefer 2 apply (blast intro: leadsETo_Trans, blast)
   368 done
   369 
   370 (**** weak ****)
   371 
   372 lemma LeadsETo_eq_leadsETo: 
   373      "A LeadsTo[CC] B =  
   374         {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC]  
   375         (reachable F Int B)}"
   376 apply (unfold LeadsETo_def)
   377 apply (blast dest: e_psp_stable2 intro: leadsETo_weaken)
   378 done
   379 
   380 (*** Introduction rules: Basis, Trans, Union ***)
   381 
   382 lemma LeadsETo_Trans:
   383      "[| F : A LeadsTo[CC] B;  F : B LeadsTo[CC] C |]  
   384       ==> F : A LeadsTo[CC] C"
   385 apply (simp add: LeadsETo_eq_leadsETo)
   386 apply (blast intro: leadsETo_Trans)
   387 done
   388 
   389 lemma LeadsETo_Union:
   390      "(!!A. A : S ==> F : A LeadsTo[CC] B) ==> F : (\<Union>S) LeadsTo[CC] B"
   391 apply (simp add: LeadsETo_def)
   392 apply (subst Int_Union)
   393 apply (blast intro: leadsETo_UN)
   394 done
   395 
   396 lemma LeadsETo_UN:
   397      "(!!i. i : I ==> F : (A i) LeadsTo[CC] B)  
   398       ==> F : (UN i:I. A i) LeadsTo[CC] B"
   399 apply (blast intro: LeadsETo_Union)
   400 done
   401 
   402 (*Binary union introduction rule*)
   403 lemma LeadsETo_Un:
   404      "[| F : A LeadsTo[CC] C; F : B LeadsTo[CC] C |]  
   405       ==> F : (A Un B) LeadsTo[CC] C"
   406   using LeadsETo_Union [of "{A, B}" F CC C] by auto
   407 
   408 (*Lets us look at the starting state*)
   409 lemma single_LeadsETo_I:
   410      "(!!s. s : A ==> F : {s} LeadsTo[CC] B) ==> F : A LeadsTo[CC] B"
   411 by (subst UN_singleton [symmetric], rule LeadsETo_UN, blast)
   412 
   413 lemma subset_imp_LeadsETo:
   414      "A <= B ==> F : A LeadsTo[CC] B"
   415 apply (simp (no_asm) add: LeadsETo_def)
   416 apply (blast intro: subset_imp_leadsETo)
   417 done
   418 
   419 lemmas empty_LeadsETo = empty_subsetI [THEN subset_imp_LeadsETo]
   420 
   421 lemma LeadsETo_weaken_R:
   422      "[| F : A LeadsTo[CC] A';  A' <= B' |] ==> F : A LeadsTo[CC] B'"
   423 apply (simp add: LeadsETo_def)
   424 apply (blast intro: leadsETo_weaken_R)
   425 done
   426 
   427 lemma LeadsETo_weaken_L:
   428      "[| F : A LeadsTo[CC] A';  B <= A |] ==> F : B LeadsTo[CC] A'"
   429 apply (simp add: LeadsETo_def)
   430 apply (blast intro: leadsETo_weaken_L)
   431 done
   432 
   433 lemma LeadsETo_weaken:
   434      "[| F : A LeadsTo[CC'] A';    
   435          B <= A;  A' <= B';  CC' <= CC |]  
   436       ==> F : B LeadsTo[CC] B'"
   437 apply (simp (no_asm_use) add: LeadsETo_def)
   438 apply (blast intro: leadsETo_weaken)
   439 done
   440 
   441 lemma LeadsETo_subset_LeadsTo: "(A LeadsTo[CC] B) <= (A LeadsTo B)"
   442 apply (unfold LeadsETo_def LeadsTo_def)
   443 apply (blast intro: leadsETo_subset_leadsTo [THEN subsetD])
   444 done
   445 
   446 (*Postcondition can be strengthened to (reachable F Int B) *)
   447 lemma reachable_ensures:
   448      "F : A ensures B ==> F : (reachable F Int A) ensures B"
   449 apply (rule stable_ensures_Int [THEN ensures_weaken_R], auto)
   450 done
   451 
   452 lemma lel_lemma:
   453      "F : A leadsTo B ==> F : (reachable F Int A) leadsTo[Pow(reachable F)] B"
   454 apply (erule leadsTo_induct)
   455   apply (blast intro: reachable_ensures)
   456  apply (blast dest: e_psp_stable2 intro: leadsETo_Trans leadsETo_weaken_L)
   457 apply (subst Int_Union)
   458 apply (blast intro: leadsETo_UN)
   459 done
   460 
   461 lemma LeadsETo_UNIV_eq_LeadsTo: "(A LeadsTo[UNIV] B) = (A LeadsTo B)"
   462 apply safe
   463 apply (erule LeadsETo_subset_LeadsTo [THEN subsetD])
   464 (*right-to-left case*)
   465 apply (unfold LeadsETo_def LeadsTo_def)
   466 apply (blast intro: lel_lemma [THEN leadsETo_weaken])
   467 done
   468 
   469 
   470 (**** EXTEND/PROJECT PROPERTIES ****)
   471 
   472 context Extend
   473 begin
   474 
   475 lemma givenBy_o_eq_extend_set:
   476      "givenBy (v o f) = extend_set h ` (givenBy v)"
   477 apply (simp add: givenBy_eq_Collect)
   478 apply (rule equalityI, best)
   479 apply blast 
   480 done
   481 
   482 lemma givenBy_eq_extend_set: "givenBy f = range (extend_set h)"
   483 by (simp add: givenBy_eq_Collect, best)
   484 
   485 lemma extend_set_givenBy_I:
   486      "D : givenBy v ==> extend_set h D : givenBy (v o f)"
   487 apply (simp (no_asm_use) add: givenBy_eq_all, blast)
   488 done
   489 
   490 lemma leadsETo_imp_extend_leadsETo:
   491      "F : A leadsTo[CC] B  
   492       ==> extend h F : (extend_set h A) leadsTo[extend_set h ` CC]  
   493                        (extend_set h B)"
   494 apply (erule leadsETo_induct)
   495   apply (force intro: subset_imp_ensures 
   496                simp add: extend_ensures extend_set_Diff_distrib [symmetric])
   497  apply (blast intro: leadsETo_Trans)
   498 apply (simp add: leadsETo_UN extend_set_Union)
   499 done
   500 
   501 
   502 (*This version's stronger in the "ensures" precondition
   503   BUT there's no ensures_weaken_L*)
   504 lemma Join_project_ensures_strong:
   505      "[| project h C G ~: transient (project_set h C Int (A-B)) |  
   506            project_set h C Int (A - B) = {};   
   507          extend h F\<squnion>G : stable C;   
   508          F\<squnion>project h C G : (project_set h C Int A) ensures B |]  
   509       ==> extend h F\<squnion>G : (C Int extend_set h A) ensures (extend_set h B)"
   510 apply (subst Int_extend_set_lemma [symmetric])
   511 apply (rule Join_project_ensures)
   512 apply (auto simp add: Int_Diff)
   513 done
   514 
   515 (*NOT WORKING.  MODIFY AS IN Project.thy
   516 lemma pld_lemma:
   517      "[| extend h F\<squnion>G : stable C;   
   518          F\<squnion>project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)`givenBy v] B;   
   519          G : preserves (v o f) |]  
   520       ==> extend h F\<squnion>G :  
   521             (C Int extend_set h (project_set h C Int A))  
   522             leadsTo[(%D. C Int extend_set h D)`givenBy v]  (extend_set h B)"
   523 apply (erule leadsETo_induct)
   524   prefer 3
   525   apply (simp del: UN_simps add: Int_UN_distrib leadsETo_UN extend_set_Union)
   526  prefer 2
   527  apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
   528 txt{*Base case is hard*}
   529 apply auto
   530 apply (force intro: leadsETo_Basis subset_imp_ensures)
   531 apply (rule leadsETo_Basis)
   532  prefer 2
   533  apply (simp add: Int_Diff Int_extend_set_lemma extend_set_Diff_distrib [symmetric])
   534 apply (rule Join_project_ensures_strong)
   535 apply (auto intro: project_stable_project_set simp add: Int_left_absorb)
   536 apply (simp (no_asm_simp) add: stable_ensures_Int [THEN ensures_weaken_R] Int_lower2 project_stable_project_set extend_stable_project_set)
   537 done
   538 
   539 lemma project_leadsETo_D_lemma:
   540      "[| extend h F\<squnion>G : stable C;   
   541          F\<squnion>project h C G :  
   542              (project_set h C Int A)  
   543              leadsTo[(%D. project_set h C Int D)`givenBy v] B;   
   544          G : preserves (v o f) |]  
   545       ==> extend h F\<squnion>G : (C Int extend_set h A)  
   546             leadsTo[(%D. C Int extend_set h D)`givenBy v] (extend_set h B)"
   547 apply (rule pld_lemma [THEN leadsETo_weaken])
   548 apply (auto simp add: split_extended_all)
   549 done
   550 
   551 lemma project_leadsETo_D:
   552      "[| F\<squnion>project h UNIV G : A leadsTo[givenBy v] B;   
   553          G : preserves (v o f) |]   
   554       ==> extend h F\<squnion>G : (extend_set h A)  
   555             leadsTo[givenBy (v o f)] (extend_set h B)"
   556 apply (cut_tac project_leadsETo_D_lemma [of _ _ UNIV], auto) 
   557 apply (erule leadsETo_givenBy)
   558 apply (rule givenBy_o_eq_extend_set [THEN equalityD2])
   559 done
   560 
   561 lemma project_LeadsETo_D:
   562      "[| F\<squnion>project h (reachable (extend h F\<squnion>G)) G  
   563              : A LeadsTo[givenBy v] B;   
   564          G : preserves (v o f) |]  
   565       ==> extend h F\<squnion>G :  
   566             (extend_set h A) LeadsTo[givenBy (v o f)] (extend_set h B)"
   567 apply (cut_tac subset_refl [THEN stable_reachable, THEN project_leadsETo_D_lemma])
   568 apply (auto simp add: LeadsETo_def)
   569  apply (erule leadsETo_mono [THEN [2] rev_subsetD])
   570  apply (blast intro: extend_set_givenBy_I)
   571 apply (simp add: project_set_reachable_extend_eq [symmetric])
   572 done
   573 
   574 lemma extending_leadsETo: 
   575      "(ALL G. extend h F ok G --> G : preserves (v o f))  
   576       ==> extending (%G. UNIV) h F  
   577                 (extend_set h A leadsTo[givenBy (v o f)] extend_set h B)  
   578                 (A leadsTo[givenBy v] B)"
   579 apply (unfold extending_def)
   580 apply (auto simp add: project_leadsETo_D)
   581 done
   582 
   583 lemma extending_LeadsETo: 
   584      "(ALL G. extend h F ok G --> G : preserves (v o f))  
   585       ==> extending (%G. reachable (extend h F\<squnion>G)) h F  
   586                 (extend_set h A LeadsTo[givenBy (v o f)] extend_set h B)  
   587                 (A LeadsTo[givenBy v]  B)"
   588 apply (unfold extending_def)
   589 apply (blast intro: project_LeadsETo_D)
   590 done
   591 *)
   592 
   593 
   594 (*** leadsETo in the precondition ***)
   595 
   596 (*Lemma for the Trans case*)
   597 lemma pli_lemma:
   598      "[| extend h F\<squnion>G : stable C;     
   599          F\<squnion>project h C G     
   600            : project_set h C Int project_set h A leadsTo project_set h B |]  
   601       ==> F\<squnion>project h C G     
   602             : project_set h C Int project_set h A leadsTo     
   603               project_set h C Int project_set h B"
   604 apply (rule psp_stable2 [THEN leadsTo_weaken_L])
   605 apply (auto simp add: project_stable_project_set extend_stable_project_set)
   606 done
   607 
   608 lemma project_leadsETo_I_lemma:
   609      "[| extend h F\<squnion>G : stable C;   
   610          extend h F\<squnion>G :  
   611            (C Int A) leadsTo[(%D. C Int D)`givenBy f]  B |]   
   612   ==> F\<squnion>project h C G   
   613     : (project_set h C Int project_set h (C Int A)) leadsTo (project_set h B)"
   614 apply (erule leadsETo_induct)
   615   prefer 3
   616   apply (simp only: Int_UN_distrib project_set_Union)
   617   apply (blast intro: leadsTo_UN)
   618  prefer 2 apply (blast intro: leadsTo_Trans pli_lemma)
   619 apply (simp add: givenBy_eq_extend_set)
   620 apply (rule leadsTo_Basis)
   621 apply (blast intro: ensures_extend_set_imp_project_ensures)
   622 done
   623 
   624 lemma project_leadsETo_I:
   625      "extend h F\<squnion>G : (extend_set h A) leadsTo[givenBy f] (extend_set h B)
   626       ==> F\<squnion>project h UNIV G : A leadsTo B"
   627 apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken], auto)
   628 done
   629 
   630 lemma project_LeadsETo_I:
   631      "extend h F\<squnion>G : (extend_set h A) LeadsTo[givenBy f] (extend_set h B) 
   632       ==> F\<squnion>project h (reachable (extend h F\<squnion>G)) G   
   633            : A LeadsTo B"
   634 apply (simp (no_asm_use) add: LeadsTo_def LeadsETo_def)
   635 apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken])
   636 apply (auto simp add: project_set_reachable_extend_eq [symmetric])
   637 done
   638 
   639 lemma projecting_leadsTo: 
   640      "projecting (%G. UNIV) h F  
   641                  (extend_set h A leadsTo[givenBy f] extend_set h B)  
   642                  (A leadsTo B)"
   643 apply (unfold projecting_def)
   644 apply (force dest: project_leadsETo_I)
   645 done
   646 
   647 lemma projecting_LeadsTo: 
   648      "projecting (%G. reachable (extend h F\<squnion>G)) h F  
   649                  (extend_set h A LeadsTo[givenBy f] extend_set h B)  
   650                  (A LeadsTo B)"
   651 apply (unfold projecting_def)
   652 apply (force dest: project_LeadsETo_I)
   653 done
   654 
   655 end
   656 
   657 end