src/HOL/UNITY/ELT.thy
author paulson
Sat Feb 08 16:05:33 2003 +0100 (2003-02-08)
changeset 13812 91713a1915ee
parent 13798 4c1a53627500
child 13819 78f5885b76a9
permissions -rw-r--r--
converting HOL/UNITY to use unconditional fairness
     1 (*  Title:      HOL/UNITY/ELT
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1999  University of Cambridge
     5 
     6 leadsTo strengthened with a specification of the allowable sets transient parts
     7 
     8 TRY INSTEAD (to get rid of the {} and to gain strong induction)
     9 
    10   elt :: "['a set set, 'a program, 'a set] => ('a set) set"
    11 
    12 inductive "elt CC F B"
    13   intros 
    14 
    15     Weaken:  "A <= B ==> A : elt CC F B"
    16 
    17     ETrans:  "[| F : A ensures A';  A-A' : CC;  A' : elt CC F B |]
    18 	      ==> A : elt CC F B"
    19 
    20     Union:  "{A. A: S} : Pow (elt CC F B) ==> (Union S) : elt CC F B"
    21 
    22   monos Pow_mono
    23 *)
    24 
    25 header{*Progress Under Allowable Sets*}
    26 
    27 theory ELT = Project:
    28 
    29 consts
    30 
    31   (*LEADS-TO constant for the inductive definition*)
    32   elt :: "['a set set, 'a program] => ('a set * 'a set) set"
    33 
    34 
    35 inductive "elt CC F"
    36  intros 
    37 
    38    Basis:  "[| F : A ensures B;  A-B : (insert {} CC) |] ==> (A,B) : elt CC F"
    39 
    40    Trans:  "[| (A,B) : elt CC F;  (B,C) : elt CC F |] ==> (A,C) : elt CC F"
    41 
    42    Union:  "ALL A: S. (A,B) : elt CC F ==> (Union S, B) : elt CC F"
    43 
    44 
    45 constdefs
    46   
    47   (*the set of all sets determined by f alone*)
    48   givenBy :: "['a => 'b] => 'a set set"
    49     "givenBy f == range (%B. f-` B)"
    50 
    51   (*visible version of the LEADS-TO relation*)
    52   leadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
    53                                         ("(3_/ leadsTo[_]/ _)" [80,0,80] 80)
    54     "leadsETo A CC B == {F. (A,B) : elt CC F}"
    55 
    56   LeadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
    57                                         ("(3_/ LeadsTo[_]/ _)" [80,0,80] 80)
    58     "LeadsETo A CC B ==
    59       {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC] B}"
    60 
    61 
    62 (*** givenBy ***)
    63 
    64 lemma givenBy_id [simp]: "givenBy id = UNIV"
    65 by (unfold givenBy_def, auto)
    66 
    67 lemma givenBy_eq_all: "(givenBy v) = {A. ALL x:A. ALL y. v x = v y --> y: A}"
    68 apply (unfold givenBy_def, safe)
    69 apply (rule_tac [2] x = "v ` ?u" in image_eqI, auto)
    70 done
    71 
    72 lemma givenByI: "(!!x y. [| x:A;  v x = v y |] ==> y: A) ==> A: givenBy v"
    73 by (subst givenBy_eq_all, blast)
    74 
    75 lemma givenByD: "[| A: givenBy v;  x:A;  v x = v y |] ==> y: A"
    76 by (unfold givenBy_def, auto)
    77 
    78 lemma empty_mem_givenBy [iff]: "{} : givenBy v"
    79 by (blast intro!: givenByI)
    80 
    81 lemma givenBy_imp_eq_Collect: "A: givenBy v ==> EX P. A = {s. P(v s)}"
    82 apply (rule_tac x = "%n. EX s. v s = n & s : A" in exI)
    83 apply (simp (no_asm_use) add: givenBy_eq_all)
    84 apply blast
    85 done
    86 
    87 lemma Collect_mem_givenBy: "{s. P(v s)} : givenBy v"
    88 by (unfold givenBy_def, best)
    89 
    90 lemma givenBy_eq_Collect: "givenBy v = {A. EX P. A = {s. P(v s)}}"
    91 by (blast intro: Collect_mem_givenBy givenBy_imp_eq_Collect)
    92 
    93 (*preserving v preserves properties given by v*)
    94 lemma preserves_givenBy_imp_stable:
    95      "[| F : preserves v;  D : givenBy v |] ==> F : stable D"
    96 by (force simp add: preserves_subset_stable [THEN subsetD] givenBy_eq_Collect)
    97 
    98 lemma givenBy_o_subset: "givenBy (w o v) <= givenBy v"
    99 apply (simp (no_asm) add: givenBy_eq_Collect)
   100 apply best 
   101 done
   102 
   103 lemma givenBy_DiffI:
   104      "[| A : givenBy v;  B : givenBy v |] ==> A-B : givenBy v"
   105 apply (simp (no_asm_use) add: givenBy_eq_Collect)
   106 apply safe
   107 apply (rule_tac x = "%z. ?R z & ~ ?Q z" in exI)
   108 apply (tactic "deepen_tac (set_cs addSIs [equalityI]) 0 1")
   109 done
   110 
   111 
   112 (** Standard leadsTo rules **)
   113 
   114 lemma leadsETo_Basis [intro]: 
   115      "[| F: A ensures B;  A-B: insert {} CC |] ==> F : A leadsTo[CC] B"
   116 apply (unfold leadsETo_def)
   117 apply (blast intro: elt.Basis)
   118 done
   119 
   120 lemma leadsETo_Trans: 
   121      "[| F : A leadsTo[CC] B;  F : B leadsTo[CC] C |] ==> F : A leadsTo[CC] C"
   122 apply (unfold leadsETo_def)
   123 apply (blast intro: elt.Trans)
   124 done
   125 
   126 
   127 (*Useful with cancellation, disjunction*)
   128 lemma leadsETo_Un_duplicate:
   129      "F : A leadsTo[CC] (A' Un A') ==> F : A leadsTo[CC] A'"
   130 apply (simp add: Un_ac)
   131 done
   132 
   133 lemma leadsETo_Un_duplicate2:
   134      "F : A leadsTo[CC] (A' Un C Un C) ==> F : A leadsTo[CC] (A' Un C)"
   135 by (simp add: Un_ac)
   136 
   137 (*The Union introduction rule as we should have liked to state it*)
   138 lemma leadsETo_Union:
   139     "(!!A. A : S ==> F : A leadsTo[CC] B) ==> F : (Union S) leadsTo[CC] B"
   140 apply (unfold leadsETo_def)
   141 apply (blast intro: elt.Union)
   142 done
   143 
   144 lemma leadsETo_UN:
   145     "(!!i. i : I ==> F : (A i) leadsTo[CC] B)  
   146      ==> F : (UN i:I. A i) leadsTo[CC] B"
   147 apply (subst Union_image_eq [symmetric])
   148 apply (blast intro: leadsETo_Union)
   149 done
   150 
   151 (*The INDUCTION rule as we should have liked to state it*)
   152 lemma leadsETo_induct:
   153   "[| F : za leadsTo[CC] zb;   
   154       !!A B. [| F : A ensures B;  A-B : insert {} CC |] ==> P A B;  
   155       !!A B C. [| F : A leadsTo[CC] B; P A B; F : B leadsTo[CC] C; P B C |]  
   156                ==> P A C;  
   157       !!B S. ALL A:S. F : A leadsTo[CC] B & P A B ==> P (Union S) B  
   158    |] ==> P za zb"
   159 apply (unfold leadsETo_def)
   160 apply (drule CollectD) 
   161 apply (erule elt.induct, blast+)
   162 done
   163 
   164 
   165 (** New facts involving leadsETo **)
   166 
   167 lemma leadsETo_mono: "CC' <= CC ==> (A leadsTo[CC'] B) <= (A leadsTo[CC] B)"
   168 apply safe
   169 apply (erule leadsETo_induct)
   170 prefer 3 apply (blast intro: leadsETo_Union)
   171 prefer 2 apply (blast intro: leadsETo_Trans)
   172 apply (blast intro: leadsETo_Basis)
   173 done
   174 
   175 lemma leadsETo_Trans_Un:
   176      "[| F : A leadsTo[CC] B;  F : B leadsTo[DD] C |]  
   177       ==> F : A leadsTo[CC Un DD] C"
   178 by (blast intro: leadsETo_mono [THEN subsetD] leadsETo_Trans)
   179 
   180 lemma leadsETo_Union_Int:
   181  "(!!A. A : S ==> F : (A Int C) leadsTo[CC] B) 
   182   ==> F : (Union S Int C) leadsTo[CC] B"
   183 apply (unfold leadsETo_def)
   184 apply (simp only: Int_Union_Union)
   185 apply (blast intro: elt.Union)
   186 done
   187 
   188 (*Binary union introduction rule*)
   189 lemma leadsETo_Un:
   190      "[| F : A leadsTo[CC] C; F : B leadsTo[CC] C |] 
   191       ==> F : (A Un B) leadsTo[CC] C"
   192 apply (subst Un_eq_Union)
   193 apply (blast intro: leadsETo_Union)
   194 done
   195 
   196 lemma single_leadsETo_I:
   197      "(!!x. x : A ==> F : {x} leadsTo[CC] B) ==> F : A leadsTo[CC] B"
   198 apply (subst UN_singleton [symmetric], rule leadsETo_UN, blast)
   199 done
   200 
   201 
   202 lemma subset_imp_leadsETo: "A<=B ==> F : A leadsTo[CC] B"
   203 by (simp add: subset_imp_ensures [THEN leadsETo_Basis] Diff_eq_empty_iff [THEN iffD2])
   204 
   205 lemmas empty_leadsETo = empty_subsetI [THEN subset_imp_leadsETo, simp]
   206 
   207 
   208 
   209 (** Weakening laws **)
   210 
   211 lemma leadsETo_weaken_R:
   212      "[| F : A leadsTo[CC] A';  A'<=B' |] ==> F : A leadsTo[CC] B'"
   213 apply (blast intro: subset_imp_leadsETo leadsETo_Trans)
   214 done
   215 
   216 lemma leadsETo_weaken_L [rule_format]:
   217      "[| F : A leadsTo[CC] A'; B<=A |] ==> F : B leadsTo[CC] A'"
   218 apply (blast intro: leadsETo_Trans subset_imp_leadsETo)
   219 done
   220 
   221 (*Distributes over binary unions*)
   222 lemma leadsETo_Un_distrib:
   223      "F : (A Un B) leadsTo[CC] C  =   
   224       (F : A leadsTo[CC] C & F : B leadsTo[CC] C)"
   225 apply (blast intro: leadsETo_Un leadsETo_weaken_L)
   226 done
   227 
   228 lemma leadsETo_UN_distrib:
   229      "F : (UN i:I. A i) leadsTo[CC] B  =   
   230       (ALL i : I. F : (A i) leadsTo[CC] B)"
   231 apply (blast intro: leadsETo_UN leadsETo_weaken_L)
   232 done
   233 
   234 lemma leadsETo_Union_distrib:
   235      "F : (Union S) leadsTo[CC] B  =  (ALL A : S. F : A leadsTo[CC] B)"
   236 apply (blast intro: leadsETo_Union leadsETo_weaken_L)
   237 done
   238 
   239 lemma leadsETo_weaken:
   240      "[| F : A leadsTo[CC'] A'; B<=A; A'<=B';  CC' <= CC |]  
   241       ==> F : B leadsTo[CC] B'"
   242 apply (drule leadsETo_mono [THEN subsetD], assumption)
   243 apply (blast del: subsetCE intro: leadsETo_weaken_R leadsETo_weaken_L leadsETo_Trans)
   244 done
   245 
   246 lemma leadsETo_givenBy:
   247      "[| F : A leadsTo[CC] A';  CC <= givenBy v |]  
   248       ==> F : A leadsTo[givenBy v] A'"
   249 by (blast intro: empty_mem_givenBy leadsETo_weaken)
   250 
   251 
   252 (*Set difference*)
   253 lemma leadsETo_Diff:
   254      "[| F : (A-B) leadsTo[CC] C; F : B leadsTo[CC] C |]  
   255       ==> F : A leadsTo[CC] C"
   256 by (blast intro: leadsETo_Un leadsETo_weaken)
   257 
   258 
   259 (*Binary union version*)
   260 lemma leadsETo_Un_Un:
   261      "[| F : A leadsTo[CC] A';  F : B leadsTo[CC] B' |]  
   262       ==> F : (A Un B) leadsTo[CC] (A' Un B')"
   263 by (blast intro: leadsETo_Un leadsETo_weaken_R)
   264 
   265 
   266 (** The cancellation law **)
   267 
   268 lemma leadsETo_cancel2:
   269      "[| F : A leadsTo[CC] (A' Un B); F : B leadsTo[CC] B' |]  
   270       ==> F : A leadsTo[CC] (A' Un B')"
   271 by (blast intro: leadsETo_Un_Un subset_imp_leadsETo leadsETo_Trans)
   272 
   273 lemma leadsETo_cancel1:
   274      "[| F : A leadsTo[CC] (B Un A'); F : B leadsTo[CC] B' |]  
   275     ==> F : A leadsTo[CC] (B' Un A')"
   276 apply (simp add: Un_commute)
   277 apply (blast intro!: leadsETo_cancel2)
   278 done
   279 
   280 lemma leadsETo_cancel_Diff1:
   281      "[| F : A leadsTo[CC] (B Un A'); F : (B-A') leadsTo[CC] B' |]  
   282     ==> F : A leadsTo[CC] (B' Un A')"
   283 apply (rule leadsETo_cancel1)
   284  prefer 2 apply assumption
   285 apply simp_all
   286 done
   287 
   288 
   289 
   290 (** PSP: Progress-Safety-Progress **)
   291 
   292 (*Special case of PSP: Misra's "stable conjunction"*)
   293 lemma e_psp_stable: 
   294    "[| F : A leadsTo[CC] A';  F : stable B;  ALL C:CC. C Int B : CC |]  
   295     ==> F : (A Int B) leadsTo[CC] (A' Int B)"
   296 apply (unfold stable_def)
   297 apply (erule leadsETo_induct)
   298 prefer 3 apply (blast intro: leadsETo_Union_Int)
   299 prefer 2 apply (blast intro: leadsETo_Trans)
   300 apply (rule leadsETo_Basis)
   301 prefer 2 apply (force simp add: Diff_Int_distrib2 [symmetric])
   302 apply (simp add: ensures_def Diff_Int_distrib2 [symmetric] Int_Un_distrib2 [symmetric])
   303 apply (blast intro: transient_strengthen constrains_Int)
   304 done
   305 
   306 lemma e_psp_stable2:
   307      "[| F : A leadsTo[CC] A'; F : stable B;  ALL C:CC. C Int B : CC |]  
   308       ==> F : (B Int A) leadsTo[CC] (B Int A')"
   309 by (simp (no_asm_simp) add: e_psp_stable Int_ac)
   310 
   311 lemma e_psp:
   312      "[| F : A leadsTo[CC] A'; F : B co B';   
   313          ALL C:CC. C Int B Int B' : CC |]  
   314       ==> F : (A Int B') leadsTo[CC] ((A' Int B) Un (B' - B))"
   315 apply (erule leadsETo_induct)
   316 prefer 3 apply (blast intro: leadsETo_Union_Int)
   317 (*Transitivity case has a delicate argument involving "cancellation"*)
   318 apply (rule_tac [2] leadsETo_Un_duplicate2)
   319 apply (erule_tac [2] leadsETo_cancel_Diff1)
   320 prefer 2
   321  apply (simp add: Int_Diff Diff_triv)
   322  apply (blast intro: leadsETo_weaken_L dest: constrains_imp_subset)
   323 (*Basis case*)
   324 apply (rule leadsETo_Basis)
   325 apply (blast intro: psp_ensures)
   326 apply (subgoal_tac "A Int B' - (Ba Int B Un (B' - B)) = (A - Ba) Int B Int B'")
   327 apply auto
   328 done
   329 
   330 lemma e_psp2:
   331      "[| F : A leadsTo[CC] A'; F : B co B';   
   332          ALL C:CC. C Int B Int B' : CC |]  
   333       ==> F : (B' Int A) leadsTo[CC] ((B Int A') Un (B' - B))"
   334 by (simp add: e_psp Int_ac)
   335 
   336 
   337 (*** Special properties involving the parameter [CC] ***)
   338 
   339 (*??IS THIS NEEDED?? or is it just an example of what's provable??*)
   340 lemma gen_leadsETo_imp_Join_leadsETo:
   341      "[| F: (A leadsTo[givenBy v] B);  G : preserves v;   
   342          F Join G : stable C |]  
   343       ==> F Join G : ((C Int A) leadsTo[(%D. C Int D) ` givenBy v] B)"
   344 apply (erule leadsETo_induct)
   345   prefer 3
   346   apply (subst Int_Union) 
   347   apply (blast intro: leadsETo_UN)
   348 prefer 2
   349  apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
   350 apply (rule leadsETo_Basis)
   351 apply (auto simp add: Diff_eq_empty_iff [THEN iffD2] Int_Diff ensures_def givenBy_eq_Collect Join_transient)
   352 prefer 3 apply (blast intro: transient_strengthen)
   353 apply (drule_tac [2] P1 = P in preserves_subset_stable [THEN subsetD])
   354 apply (drule_tac P1 = P in preserves_subset_stable [THEN subsetD])
   355 apply (unfold stable_def)
   356 apply (blast intro: constrains_Int [THEN constrains_weaken])+
   357 done
   358 
   359 (**** Relationship with traditional "leadsTo", strong & weak ****)
   360 
   361 (** strong **)
   362 
   363 lemma leadsETo_subset_leadsTo: "(A leadsTo[CC] B) <= (A leadsTo B)"
   364 apply safe
   365 apply (erule leadsETo_induct)
   366 prefer 3 apply (blast intro: leadsTo_Union)
   367 prefer 2 apply (blast intro: leadsTo_Trans, blast)
   368 done
   369 
   370 lemma leadsETo_UNIV_eq_leadsTo: "(A leadsTo[UNIV] B) = (A leadsTo B)"
   371 apply safe
   372 apply (erule leadsETo_subset_leadsTo [THEN subsetD])
   373 (*right-to-left case*)
   374 apply (erule leadsTo_induct)
   375 prefer 3 apply (blast intro: leadsETo_Union)
   376 prefer 2 apply (blast intro: leadsETo_Trans, blast)
   377 done
   378 
   379 (**** weak ****)
   380 
   381 lemma LeadsETo_eq_leadsETo: 
   382      "A LeadsTo[CC] B =  
   383         {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC]  
   384         (reachable F Int B)}"
   385 apply (unfold LeadsETo_def)
   386 apply (blast dest: e_psp_stable2 intro: leadsETo_weaken)
   387 done
   388 
   389 (*** Introduction rules: Basis, Trans, Union ***)
   390 
   391 lemma LeadsETo_Trans:
   392      "[| F : A LeadsTo[CC] B;  F : B LeadsTo[CC] C |]  
   393       ==> F : A LeadsTo[CC] C"
   394 apply (simp add: LeadsETo_eq_leadsETo)
   395 apply (blast intro: leadsETo_Trans)
   396 done
   397 
   398 lemma LeadsETo_Union:
   399      "(!!A. A : S ==> F : A LeadsTo[CC] B) ==> F : (Union S) LeadsTo[CC] B"
   400 apply (simp add: LeadsETo_def)
   401 apply (subst Int_Union)
   402 apply (blast intro: leadsETo_UN)
   403 done
   404 
   405 lemma LeadsETo_UN:
   406      "(!!i. i : I ==> F : (A i) LeadsTo[CC] B)  
   407       ==> F : (UN i:I. A i) LeadsTo[CC] B"
   408 apply (simp only: Union_image_eq [symmetric])
   409 apply (blast intro: LeadsETo_Union)
   410 done
   411 
   412 (*Binary union introduction rule*)
   413 lemma LeadsETo_Un:
   414      "[| F : A LeadsTo[CC] C; F : B LeadsTo[CC] C |]  
   415       ==> F : (A Un B) LeadsTo[CC] C"
   416 apply (subst Un_eq_Union)
   417 apply (blast intro: LeadsETo_Union)
   418 done
   419 
   420 (*Lets us look at the starting state*)
   421 lemma single_LeadsETo_I:
   422      "(!!s. s : A ==> F : {s} LeadsTo[CC] B) ==> F : A LeadsTo[CC] B"
   423 apply (subst UN_singleton [symmetric], rule LeadsETo_UN, blast)
   424 done
   425 
   426 lemma subset_imp_LeadsETo:
   427      "A <= B ==> F : A LeadsTo[CC] B"
   428 apply (simp (no_asm) add: LeadsETo_def)
   429 apply (blast intro: subset_imp_leadsETo)
   430 done
   431 
   432 lemmas empty_LeadsETo = empty_subsetI [THEN subset_imp_LeadsETo, standard]
   433 
   434 lemma LeadsETo_weaken_R [rule_format]:
   435      "[| F : A LeadsTo[CC] A';  A' <= B' |] ==> F : A LeadsTo[CC] B'"
   436 apply (simp (no_asm_use) add: LeadsETo_def)
   437 apply (blast intro: leadsETo_weaken_R)
   438 done
   439 
   440 lemma LeadsETo_weaken_L [rule_format]:
   441      "[| F : A LeadsTo[CC] A';  B <= A |] ==> F : B LeadsTo[CC] A'"
   442 apply (simp (no_asm_use) add: LeadsETo_def)
   443 apply (blast intro: leadsETo_weaken_L)
   444 done
   445 
   446 lemma LeadsETo_weaken:
   447      "[| F : A LeadsTo[CC'] A';    
   448          B <= A;  A' <= B';  CC' <= CC |]  
   449       ==> F : B LeadsTo[CC] B'"
   450 apply (simp (no_asm_use) add: LeadsETo_def)
   451 apply (blast intro: leadsETo_weaken)
   452 done
   453 
   454 lemma LeadsETo_subset_LeadsTo: "(A LeadsTo[CC] B) <= (A LeadsTo B)"
   455 apply (unfold LeadsETo_def LeadsTo_def)
   456 apply (blast intro: leadsETo_subset_leadsTo [THEN subsetD])
   457 done
   458 
   459 (*Postcondition can be strengthened to (reachable F Int B) *)
   460 lemma reachable_ensures:
   461      "F : A ensures B ==> F : (reachable F Int A) ensures B"
   462 apply (rule stable_ensures_Int [THEN ensures_weaken_R], auto)
   463 done
   464 
   465 lemma lel_lemma:
   466      "F : A leadsTo B ==> F : (reachable F Int A) leadsTo[Pow(reachable F)] B"
   467 apply (erule leadsTo_induct)
   468   apply (blast intro: reachable_ensures leadsETo_Basis)
   469  apply (blast dest: e_psp_stable2 intro: leadsETo_Trans leadsETo_weaken_L)
   470 apply (subst Int_Union)
   471 apply (blast intro: leadsETo_UN)
   472 done
   473 
   474 lemma LeadsETo_UNIV_eq_LeadsTo: "(A LeadsTo[UNIV] B) = (A LeadsTo B)"
   475 apply safe
   476 apply (erule LeadsETo_subset_LeadsTo [THEN subsetD])
   477 (*right-to-left case*)
   478 apply (unfold LeadsETo_def LeadsTo_def)
   479 apply (fast elim: lel_lemma [THEN leadsETo_weaken])
   480 done
   481 
   482 
   483 (**** EXTEND/PROJECT PROPERTIES ****)
   484 
   485 lemma (in Extend) givenBy_o_eq_extend_set: "givenBy (v o f) = extend_set h ` (givenBy v)"
   486 apply (simp (no_asm) add: givenBy_eq_Collect)
   487 apply best 
   488 done
   489 
   490 lemma (in Extend) givenBy_eq_extend_set: "givenBy f = range (extend_set h)"
   491 apply (simp (no_asm) add: givenBy_eq_Collect)
   492 apply best
   493 done
   494 
   495 lemma (in Extend) extend_set_givenBy_I:
   496      "D : givenBy v ==> extend_set h D : givenBy (v o f)"
   497 apply (simp (no_asm_use) add: givenBy_eq_all)
   498 apply blast
   499 done
   500 
   501 lemma (in Extend) leadsETo_imp_extend_leadsETo:
   502      "F : A leadsTo[CC] B  
   503       ==> extend h F : (extend_set h A) leadsTo[extend_set h ` CC]  
   504                        (extend_set h B)"
   505 apply (erule leadsETo_induct)
   506   apply (force intro: leadsETo_Basis subset_imp_ensures 
   507                simp add: extend_ensures extend_set_Diff_distrib [symmetric])
   508  apply (blast intro: leadsETo_Trans)
   509 apply (simp add: leadsETo_UN extend_set_Union)
   510 done
   511 
   512 
   513 (*This version's stronger in the "ensures" precondition
   514   BUT there's no ensures_weaken_L*)
   515 lemma (in Extend) Join_project_ensures_strong:
   516      "[| project h C G ~: transient (project_set h C Int (A-B)) |  
   517            project_set h C Int (A - B) = {};   
   518          extend h F Join G : stable C;   
   519          F Join project h C G : (project_set h C Int A) ensures B |]  
   520       ==> extend h F Join G : (C Int extend_set h A) ensures (extend_set h B)"
   521 apply (subst Int_extend_set_lemma [symmetric])
   522 apply (rule Join_project_ensures)
   523 apply (auto simp add: Int_Diff)
   524 done
   525 
   526 (*NOT WORKING.  MODIFY AS IN Project.thy
   527 lemma (in Extend) pld_lemma:
   528      "[| extend h F Join G : stable C;   
   529          F Join project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)`givenBy v] B;   
   530          G : preserves (v o f) |]  
   531       ==> extend h F Join G :  
   532             (C Int extend_set h (project_set h C Int A))  
   533             leadsTo[(%D. C Int extend_set h D)`givenBy v]  (extend_set h B)"
   534 apply (erule leadsETo_induct)
   535   prefer 3
   536   apply (simp del: UN_simps add: Int_UN_distrib leadsETo_UN extend_set_Union)
   537  prefer 2
   538  apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
   539 txt{*Base case is hard*}
   540 apply auto
   541 apply (force intro: leadsETo_Basis subset_imp_ensures)
   542 apply (rule leadsETo_Basis)
   543  prefer 2
   544  apply (simp add: Int_Diff Int_extend_set_lemma extend_set_Diff_distrib [symmetric])
   545 apply (rule Join_project_ensures_strong)
   546 apply (auto intro: project_stable_project_set simp add: Int_left_absorb)
   547 apply (simp (no_asm_simp) add: stable_ensures_Int [THEN ensures_weaken_R] Int_lower2 project_stable_project_set extend_stable_project_set)
   548 done
   549 
   550 lemma (in Extend) project_leadsETo_D_lemma:
   551      "[| extend h F Join G : stable C;   
   552          F Join project h C G :  
   553              (project_set h C Int A)  
   554              leadsTo[(%D. project_set h C Int D)`givenBy v] B;   
   555          G : preserves (v o f) |]  
   556       ==> extend h F Join G : (C Int extend_set h A)  
   557             leadsTo[(%D. C Int extend_set h D)`givenBy v] (extend_set h B)"
   558 apply (rule pld_lemma [THEN leadsETo_weaken])
   559 apply (auto simp add: split_extended_all)
   560 done
   561 
   562 lemma (in Extend) project_leadsETo_D:
   563      "[| F Join project h UNIV G : A leadsTo[givenBy v] B;   
   564          G : preserves (v o f) |]   
   565       ==> extend h F Join G : (extend_set h A)  
   566             leadsTo[givenBy (v o f)] (extend_set h B)"
   567 apply (cut_tac project_leadsETo_D_lemma [of _ _ UNIV], auto) 
   568 apply (erule leadsETo_givenBy)
   569 apply (rule givenBy_o_eq_extend_set [THEN equalityD2])
   570 done
   571 
   572 lemma (in Extend) project_LeadsETo_D:
   573      "[| F Join project h (reachable (extend h F Join G)) G  
   574              : A LeadsTo[givenBy v] B;   
   575          G : preserves (v o f) |]  
   576       ==> extend h F Join G :  
   577             (extend_set h A) LeadsTo[givenBy (v o f)] (extend_set h B)"
   578 apply (cut_tac subset_refl [THEN stable_reachable, THEN project_leadsETo_D_lemma])
   579 apply (auto simp add: LeadsETo_def)
   580  apply (erule leadsETo_mono [THEN [2] rev_subsetD])
   581  apply (blast intro: extend_set_givenBy_I)
   582 apply (simp add: project_set_reachable_extend_eq [symmetric])
   583 done
   584 
   585 lemma (in Extend) extending_leadsETo: 
   586      "(ALL G. extend h F ok G --> G : preserves (v o f))  
   587       ==> extending (%G. UNIV) h F  
   588                 (extend_set h A leadsTo[givenBy (v o f)] extend_set h B)  
   589                 (A leadsTo[givenBy v] B)"
   590 apply (unfold extending_def)
   591 apply (auto simp add: project_leadsETo_D)
   592 done
   593 
   594 lemma (in Extend) extending_LeadsETo: 
   595      "(ALL G. extend h F ok G --> G : preserves (v o f))  
   596       ==> extending (%G. reachable (extend h F Join G)) h F  
   597                 (extend_set h A LeadsTo[givenBy (v o f)] extend_set h B)  
   598                 (A LeadsTo[givenBy v]  B)"
   599 apply (unfold extending_def)
   600 apply (blast intro: project_LeadsETo_D)
   601 done
   602 *)
   603 
   604 
   605 (*** leadsETo in the precondition ***)
   606 
   607 (*Lemma for the Trans case*)
   608 lemma (in Extend) pli_lemma:
   609      "[| extend h F Join G : stable C;     
   610          F Join project h C G     
   611            : project_set h C Int project_set h A leadsTo project_set h B |]  
   612       ==> F Join project h C G     
   613             : project_set h C Int project_set h A leadsTo     
   614               project_set h C Int project_set h B"
   615 apply (rule psp_stable2 [THEN leadsTo_weaken_L])
   616 apply (auto simp add: project_stable_project_set extend_stable_project_set)
   617 done
   618 
   619 lemma (in Extend) project_leadsETo_I_lemma:
   620      "[| extend h F Join G : stable C;   
   621          extend h F Join G :  
   622            (C Int A) leadsTo[(%D. C Int D)`givenBy f]  B |]   
   623   ==> F Join project h C G   
   624     : (project_set h C Int project_set h (C Int A)) leadsTo (project_set h B)"
   625 apply (erule leadsETo_induct)
   626   prefer 3
   627   apply (simp only: Int_UN_distrib project_set_Union)
   628   apply (blast intro: leadsTo_UN)
   629  prefer 2 apply (blast intro: leadsTo_Trans pli_lemma)
   630 apply (simp add: givenBy_eq_extend_set)
   631 apply (rule leadsTo_Basis)
   632 apply (blast intro: ensures_extend_set_imp_project_ensures)
   633 done
   634 
   635 lemma (in Extend) project_leadsETo_I:
   636      "extend h F Join G : (extend_set h A) leadsTo[givenBy f] (extend_set h B)
   637       ==> F Join project h UNIV G : A leadsTo B"
   638 apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken], auto)
   639 done
   640 
   641 lemma (in Extend) project_LeadsETo_I:
   642      "extend h F Join G : (extend_set h A) LeadsTo[givenBy f] (extend_set h B) 
   643       ==> F Join project h (reachable (extend h F Join G)) G   
   644            : A LeadsTo B"
   645 apply (simp (no_asm_use) add: LeadsTo_def LeadsETo_def)
   646 apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken])
   647 apply (auto simp add: project_set_reachable_extend_eq [symmetric])
   648 done
   649 
   650 lemma (in Extend) projecting_leadsTo: 
   651      "projecting (%G. UNIV) h F  
   652                  (extend_set h A leadsTo[givenBy f] extend_set h B)  
   653                  (A leadsTo B)"
   654 apply (unfold projecting_def)
   655 apply (force dest: project_leadsETo_I)
   656 done
   657 
   658 lemma (in Extend) projecting_LeadsTo: 
   659      "projecting (%G. reachable (extend h F Join G)) h F  
   660                  (extend_set h A LeadsTo[givenBy f] extend_set h B)  
   661                  (A LeadsTo B)"
   662 apply (unfold projecting_def)
   663 apply (force dest: project_LeadsETo_I)
   664 done
   665 
   666 end