src/HOL/UNITY/ELT.thy
author wenzelm
Mon Mar 16 18:24:30 2009 +0100 (2009-03-16)
changeset 30549 d2d7874648bd
parent 26806 40b411ec05aa
child 32960 69916a850301
permissions -rw-r--r--
simplified method setup;
     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 imports Project begin
    28 
    29 inductive_set
    30   (*LEADS-TO constant for the inductive definition*)
    31   elt :: "['a set set, 'a program] => ('a set * 'a set) set"
    32   for CC :: "'a set set" and F :: "'a program"
    33  where
    34 
    35    Basis:  "[| F : A ensures B;  A-B : (insert {} CC) |] ==> (A,B) : elt CC F"
    36 
    37  | Trans:  "[| (A,B) : elt CC F;  (B,C) : elt CC F |] ==> (A,C) : elt CC F"
    38 
    39  | Union:  "ALL A: S. (A,B) : elt CC F ==> (Union S, B) : elt CC F"
    40 
    41 
    42 constdefs
    43   
    44   (*the set of all sets determined by f alone*)
    45   givenBy :: "['a => 'b] => 'a set set"
    46     "givenBy f == range (%B. f-` B)"
    47 
    48   (*visible version of the LEADS-TO relation*)
    49   leadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
    50                                         ("(3_/ leadsTo[_]/ _)" [80,0,80] 80)
    51     "leadsETo A CC B == {F. (A,B) : elt CC F}"
    52 
    53   LeadsETo :: "['a set, 'a set set, 'a set] => 'a program set"
    54                                         ("(3_/ LeadsTo[_]/ _)" [80,0,80] 80)
    55     "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 ` ?u" 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" 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 (subst Union_image_eq [symmetric])
   145 apply (blast intro: leadsETo_Union)
   146 done
   147 
   148 (*The INDUCTION rule as we should have liked to state it*)
   149 lemma leadsETo_induct:
   150   "[| F : za leadsTo[CC] zb;   
   151       !!A B. [| F : A ensures B;  A-B : insert {} CC |] ==> P A B;  
   152       !!A B C. [| F : A leadsTo[CC] B; P A B; F : B leadsTo[CC] C; P B C |]  
   153                ==> P A C;  
   154       !!B S. ALL A:S. F : A leadsTo[CC] B & P A B ==> P (Union S) B  
   155    |] ==> P za zb"
   156 apply (unfold leadsETo_def)
   157 apply (drule CollectD) 
   158 apply (erule elt.induct, blast+)
   159 done
   160 
   161 
   162 (** New facts involving leadsETo **)
   163 
   164 lemma leadsETo_mono: "CC' <= CC ==> (A leadsTo[CC'] B) <= (A leadsTo[CC] B)"
   165 apply safe
   166 apply (erule leadsETo_induct)
   167 prefer 3 apply (blast intro: leadsETo_Union)
   168 prefer 2 apply (blast intro: leadsETo_Trans)
   169 apply (blast intro: leadsETo_Basis)
   170 done
   171 
   172 lemma leadsETo_Trans_Un:
   173      "[| F : A leadsTo[CC] B;  F : B leadsTo[DD] C |]  
   174       ==> F : A leadsTo[CC Un DD] C"
   175 by (blast intro: leadsETo_mono [THEN subsetD] leadsETo_Trans)
   176 
   177 lemma leadsETo_Union_Int:
   178  "(!!A. A : S ==> F : (A Int C) leadsTo[CC] B) 
   179   ==> F : (Union S Int C) leadsTo[CC] B"
   180 apply (unfold leadsETo_def)
   181 apply (simp only: Int_Union_Union)
   182 apply (blast intro: elt.Union)
   183 done
   184 
   185 (*Binary union introduction rule*)
   186 lemma leadsETo_Un:
   187      "[| F : A leadsTo[CC] C; F : B leadsTo[CC] C |] 
   188       ==> F : (A Un B) leadsTo[CC] C"
   189 apply (subst Un_eq_Union)
   190 apply (blast intro: leadsETo_Union)
   191 done
   192 
   193 lemma single_leadsETo_I:
   194      "(!!x. x : A ==> F : {x} leadsTo[CC] B) ==> F : A leadsTo[CC] B"
   195 by (subst UN_singleton [symmetric], rule leadsETo_UN, blast)
   196 
   197 
   198 lemma subset_imp_leadsETo: "A<=B ==> F : A leadsTo[CC] B"
   199 by (simp add: subset_imp_ensures [THEN leadsETo_Basis] 
   200               Diff_eq_empty_iff [THEN iffD2])
   201 
   202 lemmas empty_leadsETo = empty_subsetI [THEN subset_imp_leadsETo, simp]
   203 
   204 
   205 
   206 (** Weakening laws **)
   207 
   208 lemma leadsETo_weaken_R:
   209      "[| F : A leadsTo[CC] A';  A'<=B' |] ==> F : A leadsTo[CC] B'"
   210 by (blast intro: subset_imp_leadsETo leadsETo_Trans)
   211 
   212 lemma leadsETo_weaken_L [rule_format]:
   213      "[| F : A leadsTo[CC] A'; B<=A |] ==> F : B leadsTo[CC] A'"
   214 by (blast intro: leadsETo_Trans subset_imp_leadsETo)
   215 
   216 (*Distributes over binary unions*)
   217 lemma leadsETo_Un_distrib:
   218      "F : (A Un B) leadsTo[CC] C  =   
   219       (F : A leadsTo[CC] C & F : B leadsTo[CC] C)"
   220 by (blast intro: leadsETo_Un leadsETo_weaken_L)
   221 
   222 lemma leadsETo_UN_distrib:
   223      "F : (UN i:I. A i) leadsTo[CC] B  =   
   224       (ALL i : I. F : (A i) leadsTo[CC] B)"
   225 by (blast intro: leadsETo_UN leadsETo_weaken_L)
   226 
   227 lemma leadsETo_Union_distrib:
   228      "F : (Union S) leadsTo[CC] B  =  (ALL A : S. F : A leadsTo[CC] B)"
   229 by (blast intro: leadsETo_Union leadsETo_weaken_L)
   230 
   231 lemma leadsETo_weaken:
   232      "[| F : A leadsTo[CC'] A'; B<=A; A'<=B';  CC' <= CC |]  
   233       ==> F : B leadsTo[CC] B'"
   234 apply (drule leadsETo_mono [THEN subsetD], assumption)
   235 apply (blast del: subsetCE 
   236              intro: leadsETo_weaken_R leadsETo_weaken_L leadsETo_Trans)
   237 done
   238 
   239 lemma leadsETo_givenBy:
   240      "[| F : A leadsTo[CC] A';  CC <= givenBy v |]  
   241       ==> F : A leadsTo[givenBy v] A'"
   242 by (blast intro: empty_mem_givenBy leadsETo_weaken)
   243 
   244 
   245 (*Set difference*)
   246 lemma leadsETo_Diff:
   247      "[| F : (A-B) leadsTo[CC] C; F : B leadsTo[CC] C |]  
   248       ==> F : A leadsTo[CC] C"
   249 by (blast intro: leadsETo_Un leadsETo_weaken)
   250 
   251 
   252 (*Binary union version*)
   253 lemma leadsETo_Un_Un:
   254      "[| F : A leadsTo[CC] A';  F : B leadsTo[CC] B' |]  
   255       ==> F : (A Un B) leadsTo[CC] (A' Un B')"
   256 by (blast intro: leadsETo_Un leadsETo_weaken_R)
   257 
   258 
   259 (** The cancellation law **)
   260 
   261 lemma leadsETo_cancel2:
   262      "[| F : A leadsTo[CC] (A' Un B); F : B leadsTo[CC] B' |]  
   263       ==> F : A leadsTo[CC] (A' Un B')"
   264 by (blast intro: leadsETo_Un_Un subset_imp_leadsETo leadsETo_Trans)
   265 
   266 lemma leadsETo_cancel1:
   267      "[| F : A leadsTo[CC] (B Un A'); F : B leadsTo[CC] B' |]  
   268     ==> F : A leadsTo[CC] (B' Un A')"
   269 apply (simp add: Un_commute)
   270 apply (blast intro!: leadsETo_cancel2)
   271 done
   272 
   273 lemma leadsETo_cancel_Diff1:
   274      "[| F : A leadsTo[CC] (B Un A'); F : (B-A') leadsTo[CC] B' |]  
   275     ==> F : A leadsTo[CC] (B' Un A')"
   276 apply (rule leadsETo_cancel1)
   277  prefer 2 apply assumption
   278 apply simp_all
   279 done
   280 
   281 
   282 (** PSP: Progress-Safety-Progress **)
   283 
   284 (*Special case of PSP: Misra's "stable conjunction"*)
   285 lemma e_psp_stable: 
   286    "[| F : A leadsTo[CC] A';  F : stable B;  ALL C:CC. C Int B : CC |]  
   287     ==> F : (A Int B) leadsTo[CC] (A' Int B)"
   288 apply (unfold stable_def)
   289 apply (erule leadsETo_induct)
   290 prefer 3 apply (blast intro: leadsETo_Union_Int)
   291 prefer 2 apply (blast intro: leadsETo_Trans)
   292 apply (rule leadsETo_Basis)
   293 prefer 2 apply (force simp add: Diff_Int_distrib2 [symmetric])
   294 apply (simp add: ensures_def Diff_Int_distrib2 [symmetric] 
   295                  Int_Un_distrib2 [symmetric])
   296 apply (blast intro: transient_strengthen constrains_Int)
   297 done
   298 
   299 lemma e_psp_stable2:
   300      "[| F : A leadsTo[CC] A'; F : stable B;  ALL C:CC. C Int B : CC |]  
   301       ==> F : (B Int A) leadsTo[CC] (B Int A')"
   302 by (simp (no_asm_simp) add: e_psp_stable Int_ac)
   303 
   304 lemma e_psp:
   305      "[| F : A leadsTo[CC] A'; F : B co B';   
   306          ALL C:CC. C Int B Int B' : CC |]  
   307       ==> F : (A Int B') leadsTo[CC] ((A' Int B) Un (B' - B))"
   308 apply (erule leadsETo_induct)
   309 prefer 3 apply (blast intro: leadsETo_Union_Int)
   310 (*Transitivity case has a delicate argument involving "cancellation"*)
   311 apply (rule_tac [2] leadsETo_Un_duplicate2)
   312 apply (erule_tac [2] leadsETo_cancel_Diff1)
   313 prefer 2
   314  apply (simp add: Int_Diff Diff_triv)
   315  apply (blast intro: leadsETo_weaken_L dest: constrains_imp_subset)
   316 (*Basis case*)
   317 apply (rule leadsETo_Basis)
   318 apply (blast intro: psp_ensures)
   319 apply (subgoal_tac "A Int B' - (Ba Int B Un (B' - B)) = (A - Ba) Int B Int B'")
   320 apply auto
   321 done
   322 
   323 lemma e_psp2:
   324      "[| F : A leadsTo[CC] A'; F : B co B';   
   325          ALL C:CC. C Int B Int B' : CC |]  
   326       ==> F : (B' Int A) leadsTo[CC] ((B Int A') Un (B' - B))"
   327 by (simp add: e_psp Int_ac)
   328 
   329 
   330 (*** Special properties involving the parameter [CC] ***)
   331 
   332 (*??IS THIS NEEDED?? or is it just an example of what's provable??*)
   333 lemma gen_leadsETo_imp_Join_leadsETo:
   334      "[| F: (A leadsTo[givenBy v] B);  G : preserves v;   
   335          F\<squnion>G : stable C |]  
   336       ==> F\<squnion>G : ((C Int A) leadsTo[(%D. C Int D) ` givenBy v] B)"
   337 apply (erule leadsETo_induct)
   338   prefer 3
   339   apply (subst Int_Union) 
   340   apply (blast intro: leadsETo_UN)
   341 prefer 2
   342  apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
   343 apply (rule leadsETo_Basis)
   344 apply (auto simp add: Diff_eq_empty_iff [THEN iffD2] 
   345                       Int_Diff ensures_def givenBy_eq_Collect Join_transient)
   346 prefer 3 apply (blast intro: transient_strengthen)
   347 apply (drule_tac [2] P1 = P in preserves_subset_stable [THEN subsetD])
   348 apply (drule_tac P1 = P in preserves_subset_stable [THEN subsetD])
   349 apply (unfold stable_def)
   350 apply (blast intro: constrains_Int [THEN constrains_weaken])+
   351 done
   352 
   353 (**** Relationship with traditional "leadsTo", strong & weak ****)
   354 
   355 (** strong **)
   356 
   357 lemma leadsETo_subset_leadsTo: "(A leadsTo[CC] B) <= (A leadsTo B)"
   358 apply safe
   359 apply (erule leadsETo_induct)
   360   prefer 3 apply (blast intro: leadsTo_Union)
   361  prefer 2 apply (blast intro: leadsTo_Trans, blast)
   362 done
   363 
   364 lemma leadsETo_UNIV_eq_leadsTo: "(A leadsTo[UNIV] B) = (A leadsTo B)"
   365 apply safe
   366 apply (erule leadsETo_subset_leadsTo [THEN subsetD])
   367 (*right-to-left case*)
   368 apply (erule leadsTo_induct)
   369   prefer 3 apply (blast intro: leadsETo_Union)
   370  prefer 2 apply (blast intro: leadsETo_Trans, blast)
   371 done
   372 
   373 (**** weak ****)
   374 
   375 lemma LeadsETo_eq_leadsETo: 
   376      "A LeadsTo[CC] B =  
   377         {F. F : (reachable F Int A) leadsTo[(%C. reachable F Int C) ` CC]  
   378         (reachable F Int B)}"
   379 apply (unfold LeadsETo_def)
   380 apply (blast dest: e_psp_stable2 intro: leadsETo_weaken)
   381 done
   382 
   383 (*** Introduction rules: Basis, Trans, Union ***)
   384 
   385 lemma LeadsETo_Trans:
   386      "[| F : A LeadsTo[CC] B;  F : B LeadsTo[CC] C |]  
   387       ==> F : A LeadsTo[CC] C"
   388 apply (simp add: LeadsETo_eq_leadsETo)
   389 apply (blast intro: leadsETo_Trans)
   390 done
   391 
   392 lemma LeadsETo_Union:
   393      "(!!A. A : S ==> F : A LeadsTo[CC] B) ==> F : (Union S) LeadsTo[CC] B"
   394 apply (simp add: LeadsETo_def)
   395 apply (subst Int_Union)
   396 apply (blast intro: leadsETo_UN)
   397 done
   398 
   399 lemma LeadsETo_UN:
   400      "(!!i. i : I ==> F : (A i) LeadsTo[CC] B)  
   401       ==> F : (UN i:I. A i) LeadsTo[CC] B"
   402 apply (simp only: Union_image_eq [symmetric])
   403 apply (blast intro: LeadsETo_Union)
   404 done
   405 
   406 (*Binary union introduction rule*)
   407 lemma LeadsETo_Un:
   408      "[| F : A LeadsTo[CC] C; F : B LeadsTo[CC] C |]  
   409       ==> F : (A Un B) LeadsTo[CC] C"
   410 apply (subst Un_eq_Union)
   411 apply (blast intro: LeadsETo_Union)
   412 done
   413 
   414 (*Lets us look at the starting state*)
   415 lemma single_LeadsETo_I:
   416      "(!!s. s : A ==> F : {s} LeadsTo[CC] B) ==> F : A LeadsTo[CC] B"
   417 by (subst UN_singleton [symmetric], rule LeadsETo_UN, blast)
   418 
   419 lemma subset_imp_LeadsETo:
   420      "A <= B ==> F : A LeadsTo[CC] B"
   421 apply (simp (no_asm) add: LeadsETo_def)
   422 apply (blast intro: subset_imp_leadsETo)
   423 done
   424 
   425 lemmas empty_LeadsETo = empty_subsetI [THEN subset_imp_LeadsETo, standard]
   426 
   427 lemma LeadsETo_weaken_R [rule_format]:
   428      "[| F : A LeadsTo[CC] A';  A' <= B' |] ==> F : A LeadsTo[CC] B'"
   429 apply (simp (no_asm_use) add: LeadsETo_def)
   430 apply (blast intro: leadsETo_weaken_R)
   431 done
   432 
   433 lemma LeadsETo_weaken_L [rule_format]:
   434      "[| F : A LeadsTo[CC] A';  B <= A |] ==> F : B LeadsTo[CC] A'"
   435 apply (simp (no_asm_use) add: LeadsETo_def)
   436 apply (blast intro: leadsETo_weaken_L)
   437 done
   438 
   439 lemma LeadsETo_weaken:
   440      "[| F : A LeadsTo[CC'] A';    
   441          B <= A;  A' <= B';  CC' <= CC |]  
   442       ==> F : B LeadsTo[CC] B'"
   443 apply (simp (no_asm_use) add: LeadsETo_def)
   444 apply (blast intro: leadsETo_weaken)
   445 done
   446 
   447 lemma LeadsETo_subset_LeadsTo: "(A LeadsTo[CC] B) <= (A LeadsTo B)"
   448 apply (unfold LeadsETo_def LeadsTo_def)
   449 apply (blast intro: leadsETo_subset_leadsTo [THEN subsetD])
   450 done
   451 
   452 (*Postcondition can be strengthened to (reachable F Int B) *)
   453 lemma reachable_ensures:
   454      "F : A ensures B ==> F : (reachable F Int A) ensures B"
   455 apply (rule stable_ensures_Int [THEN ensures_weaken_R], auto)
   456 done
   457 
   458 lemma lel_lemma:
   459      "F : A leadsTo B ==> F : (reachable F Int A) leadsTo[Pow(reachable F)] B"
   460 apply (erule leadsTo_induct)
   461   apply (blast intro: reachable_ensures leadsETo_Basis)
   462  apply (blast dest: e_psp_stable2 intro: leadsETo_Trans leadsETo_weaken_L)
   463 apply (subst Int_Union)
   464 apply (blast intro: leadsETo_UN)
   465 done
   466 
   467 lemma LeadsETo_UNIV_eq_LeadsTo: "(A LeadsTo[UNIV] B) = (A LeadsTo B)"
   468 apply safe
   469 apply (erule LeadsETo_subset_LeadsTo [THEN subsetD])
   470 (*right-to-left case*)
   471 apply (unfold LeadsETo_def LeadsTo_def)
   472 apply (blast intro: lel_lemma [THEN leadsETo_weaken])
   473 done
   474 
   475 
   476 (**** EXTEND/PROJECT PROPERTIES ****)
   477 
   478 lemma (in Extend) givenBy_o_eq_extend_set:
   479      "givenBy (v o f) = extend_set h ` (givenBy v)"
   480 apply (simp add: givenBy_eq_Collect)
   481 apply (rule equalityI, best)
   482 apply blast 
   483 done
   484 
   485 lemma (in Extend) givenBy_eq_extend_set: "givenBy f = range (extend_set h)"
   486 by (simp add: givenBy_eq_Collect, best)
   487 
   488 lemma (in Extend) extend_set_givenBy_I:
   489      "D : givenBy v ==> extend_set h D : givenBy (v o f)"
   490 apply (simp (no_asm_use) add: givenBy_eq_all, blast)
   491 done
   492 
   493 lemma (in Extend) leadsETo_imp_extend_leadsETo:
   494      "F : A leadsTo[CC] B  
   495       ==> extend h F : (extend_set h A) leadsTo[extend_set h ` CC]  
   496                        (extend_set h B)"
   497 apply (erule leadsETo_induct)
   498   apply (force intro: leadsETo_Basis subset_imp_ensures 
   499                simp add: extend_ensures extend_set_Diff_distrib [symmetric])
   500  apply (blast intro: leadsETo_Trans)
   501 apply (simp add: leadsETo_UN extend_set_Union)
   502 done
   503 
   504 
   505 (*This version's stronger in the "ensures" precondition
   506   BUT there's no ensures_weaken_L*)
   507 lemma (in Extend) Join_project_ensures_strong:
   508      "[| project h C G ~: transient (project_set h C Int (A-B)) |  
   509            project_set h C Int (A - B) = {};   
   510          extend h F\<squnion>G : stable C;   
   511          F\<squnion>project h C G : (project_set h C Int A) ensures B |]  
   512       ==> extend h F\<squnion>G : (C Int extend_set h A) ensures (extend_set h B)"
   513 apply (subst Int_extend_set_lemma [symmetric])
   514 apply (rule Join_project_ensures)
   515 apply (auto simp add: Int_Diff)
   516 done
   517 
   518 (*NOT WORKING.  MODIFY AS IN Project.thy
   519 lemma (in Extend) pld_lemma:
   520      "[| extend h F\<squnion>G : stable C;   
   521          F\<squnion>project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)`givenBy v] B;   
   522          G : preserves (v o f) |]  
   523       ==> extend h F\<squnion>G :  
   524             (C Int extend_set h (project_set h C Int A))  
   525             leadsTo[(%D. C Int extend_set h D)`givenBy v]  (extend_set h B)"
   526 apply (erule leadsETo_induct)
   527   prefer 3
   528   apply (simp del: UN_simps add: Int_UN_distrib leadsETo_UN extend_set_Union)
   529  prefer 2
   530  apply (blast intro: e_psp_stable2 [THEN leadsETo_weaken_L] leadsETo_Trans)
   531 txt{*Base case is hard*}
   532 apply auto
   533 apply (force intro: leadsETo_Basis subset_imp_ensures)
   534 apply (rule leadsETo_Basis)
   535  prefer 2
   536  apply (simp add: Int_Diff Int_extend_set_lemma extend_set_Diff_distrib [symmetric])
   537 apply (rule Join_project_ensures_strong)
   538 apply (auto intro: project_stable_project_set simp add: Int_left_absorb)
   539 apply (simp (no_asm_simp) add: stable_ensures_Int [THEN ensures_weaken_R] Int_lower2 project_stable_project_set extend_stable_project_set)
   540 done
   541 
   542 lemma (in Extend) project_leadsETo_D_lemma:
   543      "[| extend h F\<squnion>G : stable C;   
   544          F\<squnion>project h C G :  
   545              (project_set h C Int A)  
   546              leadsTo[(%D. project_set h C Int D)`givenBy v] B;   
   547          G : preserves (v o f) |]  
   548       ==> extend h F\<squnion>G : (C Int extend_set h A)  
   549             leadsTo[(%D. C Int extend_set h D)`givenBy v] (extend_set h B)"
   550 apply (rule pld_lemma [THEN leadsETo_weaken])
   551 apply (auto simp add: split_extended_all)
   552 done
   553 
   554 lemma (in Extend) project_leadsETo_D:
   555      "[| F\<squnion>project h UNIV G : A leadsTo[givenBy v] B;   
   556          G : preserves (v o f) |]   
   557       ==> extend h F\<squnion>G : (extend_set h A)  
   558             leadsTo[givenBy (v o f)] (extend_set h B)"
   559 apply (cut_tac project_leadsETo_D_lemma [of _ _ UNIV], auto) 
   560 apply (erule leadsETo_givenBy)
   561 apply (rule givenBy_o_eq_extend_set [THEN equalityD2])
   562 done
   563 
   564 lemma (in Extend) project_LeadsETo_D:
   565      "[| F\<squnion>project h (reachable (extend h F\<squnion>G)) G  
   566              : A LeadsTo[givenBy v] B;   
   567          G : preserves (v o f) |]  
   568       ==> extend h F\<squnion>G :  
   569             (extend_set h A) LeadsTo[givenBy (v o f)] (extend_set h B)"
   570 apply (cut_tac subset_refl [THEN stable_reachable, THEN project_leadsETo_D_lemma])
   571 apply (auto simp add: LeadsETo_def)
   572  apply (erule leadsETo_mono [THEN [2] rev_subsetD])
   573  apply (blast intro: extend_set_givenBy_I)
   574 apply (simp add: project_set_reachable_extend_eq [symmetric])
   575 done
   576 
   577 lemma (in Extend) extending_leadsETo: 
   578      "(ALL G. extend h F ok G --> G : preserves (v o f))  
   579       ==> extending (%G. UNIV) h F  
   580                 (extend_set h A leadsTo[givenBy (v o f)] extend_set h B)  
   581                 (A leadsTo[givenBy v] B)"
   582 apply (unfold extending_def)
   583 apply (auto simp add: project_leadsETo_D)
   584 done
   585 
   586 lemma (in Extend) extending_LeadsETo: 
   587      "(ALL G. extend h F ok G --> G : preserves (v o f))  
   588       ==> extending (%G. reachable (extend h F\<squnion>G)) h F  
   589                 (extend_set h A LeadsTo[givenBy (v o f)] extend_set h B)  
   590                 (A LeadsTo[givenBy v]  B)"
   591 apply (unfold extending_def)
   592 apply (blast intro: project_LeadsETo_D)
   593 done
   594 *)
   595 
   596 
   597 (*** leadsETo in the precondition ***)
   598 
   599 (*Lemma for the Trans case*)
   600 lemma (in Extend) pli_lemma:
   601      "[| extend h F\<squnion>G : stable C;     
   602          F\<squnion>project h C G     
   603            : project_set h C Int project_set h A leadsTo project_set h B |]  
   604       ==> F\<squnion>project h C G     
   605             : project_set h C Int project_set h A leadsTo     
   606               project_set h C Int project_set h B"
   607 apply (rule psp_stable2 [THEN leadsTo_weaken_L])
   608 apply (auto simp add: project_stable_project_set extend_stable_project_set)
   609 done
   610 
   611 lemma (in Extend) project_leadsETo_I_lemma:
   612      "[| extend h F\<squnion>G : stable C;   
   613          extend h F\<squnion>G :  
   614            (C Int A) leadsTo[(%D. C Int D)`givenBy f]  B |]   
   615   ==> F\<squnion>project h C G   
   616     : (project_set h C Int project_set h (C Int A)) leadsTo (project_set h B)"
   617 apply (erule leadsETo_induct)
   618   prefer 3
   619   apply (simp only: Int_UN_distrib project_set_Union)
   620   apply (blast intro: leadsTo_UN)
   621  prefer 2 apply (blast intro: leadsTo_Trans pli_lemma)
   622 apply (simp add: givenBy_eq_extend_set)
   623 apply (rule leadsTo_Basis)
   624 apply (blast intro: ensures_extend_set_imp_project_ensures)
   625 done
   626 
   627 lemma (in Extend) project_leadsETo_I:
   628      "extend h F\<squnion>G : (extend_set h A) leadsTo[givenBy f] (extend_set h B)
   629       ==> F\<squnion>project h UNIV G : A leadsTo B"
   630 apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken], auto)
   631 done
   632 
   633 lemma (in Extend) project_LeadsETo_I:
   634      "extend h F\<squnion>G : (extend_set h A) LeadsTo[givenBy f] (extend_set h B) 
   635       ==> F\<squnion>project h (reachable (extend h F\<squnion>G)) G   
   636            : A LeadsTo B"
   637 apply (simp (no_asm_use) add: LeadsTo_def LeadsETo_def)
   638 apply (rule project_leadsETo_I_lemma [THEN leadsTo_weaken])
   639 apply (auto simp add: project_set_reachable_extend_eq [symmetric])
   640 done
   641 
   642 lemma (in Extend) projecting_leadsTo: 
   643      "projecting (%G. UNIV) h F  
   644                  (extend_set h A leadsTo[givenBy f] extend_set h B)  
   645                  (A leadsTo B)"
   646 apply (unfold projecting_def)
   647 apply (force dest: project_leadsETo_I)
   648 done
   649 
   650 lemma (in Extend) projecting_LeadsTo: 
   651      "projecting (%G. reachable (extend h F\<squnion>G)) 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 end