src/HOL/UNITY/Simple/Lift.thy
author paulson
Sat Feb 08 16:05:33 2003 +0100 (2003-02-08)
changeset 13812 91713a1915ee
parent 13806 fd40c9d9076b
child 14378 69c4d5997669
permissions -rw-r--r--
converting HOL/UNITY to use unconditional fairness
     1 (*  Title:      HOL/UNITY/Lift.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1998  University of Cambridge
     5 
     6 The Lift-Control Example
     7 *)
     8 
     9 theory Lift = UNITY_Main:
    10 
    11 record state =
    12   floor :: "int"	    (*current position of the lift*)
    13   "open" :: "bool"	    (*whether the door is opened at floor*)
    14   stop  :: "bool"	    (*whether the lift is stopped at floor*)
    15   req   :: "int set"	    (*for each floor, whether the lift is requested*)
    16   up    :: "bool"	    (*current direction of movement*)
    17   move  :: "bool"	    (*whether moving takes precedence over opening*)
    18 
    19 consts
    20   Min :: "int"       (*least and greatest floors*)
    21   Max :: "int"       (*least and greatest floors*)
    22 
    23 axioms
    24   Min_le_Max [iff]: "Min \<le> Max"
    25   
    26 constdefs
    27   
    28   (** Abbreviations: the "always" part **)
    29   
    30   above :: "state set"
    31     "above == {s. \<exists>i. floor s < i & i \<le> Max & i \<in> req s}"
    32 
    33   below :: "state set"
    34     "below == {s. \<exists>i. Min \<le> i & i < floor s & i \<in> req s}"
    35 
    36   queueing :: "state set"
    37     "queueing == above \<union> below"
    38 
    39   goingup :: "state set"
    40     "goingup   == above \<inter> ({s. up s}  \<union> -below)"
    41 
    42   goingdown :: "state set"
    43     "goingdown == below \<inter> ({s. ~ up s} \<union> -above)"
    44 
    45   ready :: "state set"
    46     "ready == {s. stop s & ~ open s & move s}"
    47  
    48   (** Further abbreviations **)
    49 
    50   moving :: "state set"
    51     "moving ==  {s. ~ stop s & ~ open s}"
    52 
    53   stopped :: "state set"
    54     "stopped == {s. stop s  & ~ open s & ~ move s}"
    55 
    56   opened :: "state set"
    57     "opened ==  {s. stop s  &  open s  &  move s}"
    58 
    59   closed :: "state set"  (*but this is the same as ready!!*)
    60     "closed ==  {s. stop s  & ~ open s &  move s}"
    61 
    62   atFloor :: "int => state set"
    63     "atFloor n ==  {s. floor s = n}"
    64 
    65   Req :: "int => state set"
    66     "Req n ==  {s. n \<in> req s}"
    67 
    68 
    69   
    70   (** The program **)
    71   
    72   request_act :: "(state*state) set"
    73     "request_act == {(s,s'). s' = s (|stop:=True, move:=False|)
    74 		                  & ~ stop s & floor s \<in> req s}"
    75 
    76   open_act :: "(state*state) set"
    77     "open_act ==
    78          {(s,s'). s' = s (|open :=True,
    79 			   req  := req s - {floor s},
    80 			   move := True|)
    81 		       & stop s & ~ open s & floor s \<in> req s
    82 	               & ~(move s & s \<in> queueing)}"
    83 
    84   close_act :: "(state*state) set"
    85     "close_act == {(s,s'). s' = s (|open := False|) & open s}"
    86 
    87   req_up :: "(state*state) set"
    88     "req_up ==
    89          {(s,s'). s' = s (|stop  :=False,
    90 			   floor := floor s + 1,
    91 			   up    := True|)
    92 		       & s \<in> (ready \<inter> goingup)}"
    93 
    94   req_down :: "(state*state) set"
    95     "req_down ==
    96          {(s,s'). s' = s (|stop  :=False,
    97 			   floor := floor s - 1,
    98 			   up    := False|)
    99 		       & s \<in> (ready \<inter> goingdown)}"
   100 
   101   move_up :: "(state*state) set"
   102     "move_up ==
   103          {(s,s'). s' = s (|floor := floor s + 1|)
   104 		       & ~ stop s & up s & floor s \<notin> req s}"
   105 
   106   move_down :: "(state*state) set"
   107     "move_down ==
   108          {(s,s'). s' = s (|floor := floor s - 1|)
   109 		       & ~ stop s & ~ up s & floor s \<notin> req s}"
   110 
   111   (*This action is omitted from prior treatments, which therefore are
   112     unrealistic: nobody asks the lift to do anything!  But adding this
   113     action invalidates many of the existing progress arguments: various
   114     "ensures" properties fail.*)
   115   button_press  :: "(state*state) set"
   116     "button_press ==
   117          {(s,s'). \<exists>n. s' = s (|req := insert n (req s)|)
   118 		        & Min \<le> n & n \<le> Max}"
   119 
   120 
   121   Lift :: "state program"
   122     (*for the moment, we OMIT button_press*)
   123     "Lift == mk_total_program
   124                 ({s. floor s = Min & ~ up s & move s & stop s &
   125 		          ~ open s & req s = {}},
   126 		 {request_act, open_act, close_act,
   127  		  req_up, req_down, move_up, move_down},
   128 		 UNIV)"
   129 
   130 
   131   (** Invariants **)
   132 
   133   bounded :: "state set"
   134     "bounded == {s. Min \<le> floor s & floor s \<le> Max}"
   135 
   136   open_stop :: "state set"
   137     "open_stop == {s. open s --> stop s}"
   138   
   139   open_move :: "state set"
   140     "open_move == {s. open s --> move s}"
   141   
   142   stop_floor :: "state set"
   143     "stop_floor == {s. stop s & ~ move s --> floor s \<in> req s}"
   144   
   145   moving_up :: "state set"
   146     "moving_up == {s. ~ stop s & up s -->
   147                    (\<exists>f. floor s \<le> f & f \<le> Max & f \<in> req s)}"
   148   
   149   moving_down :: "state set"
   150     "moving_down == {s. ~ stop s & ~ up s -->
   151                      (\<exists>f. Min \<le> f & f \<le> floor s & f \<in> req s)}"
   152   
   153   metric :: "[int,state] => int"
   154     "metric ==
   155        %n s. if floor s < n then (if up s then n - floor s
   156 			          else (floor s - Min) + (n-Min))
   157              else
   158              if n < floor s then (if up s then (Max - floor s) + (Max-n)
   159 		                  else floor s - n)
   160              else 0"
   161 
   162 locale Floor =
   163   fixes n
   164   assumes Min_le_n [iff]: "Min \<le> n"
   165       and n_le_Max [iff]: "n \<le> Max"
   166 
   167 lemma not_mem_distinct: "[| x \<notin> A;  y \<in> A |] ==> x \<noteq> y"
   168 by blast
   169 
   170 
   171 declare Lift_def [THEN def_prg_Init, simp]
   172 
   173 declare request_act_def [THEN def_act_simp, simp]
   174 declare open_act_def [THEN def_act_simp, simp]
   175 declare close_act_def [THEN def_act_simp, simp]
   176 declare req_up_def [THEN def_act_simp, simp]
   177 declare req_down_def [THEN def_act_simp, simp]
   178 declare move_up_def [THEN def_act_simp, simp]
   179 declare move_down_def [THEN def_act_simp, simp]
   180 declare button_press_def [THEN def_act_simp, simp]
   181 
   182 (*The ALWAYS properties*)
   183 declare above_def [THEN def_set_simp, simp]
   184 declare below_def [THEN def_set_simp, simp]
   185 declare queueing_def [THEN def_set_simp, simp]
   186 declare goingup_def [THEN def_set_simp, simp]
   187 declare goingdown_def [THEN def_set_simp, simp]
   188 declare ready_def [THEN def_set_simp, simp]
   189 
   190 (*Basic definitions*)
   191 declare bounded_def [simp] 
   192         open_stop_def [simp] 
   193         open_move_def [simp] 
   194         stop_floor_def [simp]
   195         moving_up_def [simp]
   196         moving_down_def [simp]
   197 
   198 lemma open_stop: "Lift \<in> Always open_stop"
   199 apply (rule AlwaysI, force) 
   200 apply (unfold Lift_def, constrains)
   201 done
   202 
   203 lemma stop_floor: "Lift \<in> Always stop_floor"
   204 apply (rule AlwaysI, force) 
   205 apply (unfold Lift_def, constrains)
   206 done
   207 
   208 (*This one needs open_stop, which was proved above*)
   209 lemma open_move: "Lift \<in> Always open_move"
   210 apply (cut_tac open_stop)
   211 apply (rule AlwaysI, force) 
   212 apply (unfold Lift_def, constrains)
   213 done
   214 
   215 lemma moving_up: "Lift \<in> Always moving_up"
   216 apply (rule AlwaysI, force) 
   217 apply (unfold Lift_def, constrains)
   218 apply (auto dest: zle_imp_zless_or_eq simp add: add1_zle_eq)
   219 done
   220 
   221 lemma moving_down: "Lift \<in> Always moving_down"
   222 apply (rule AlwaysI, force) 
   223 apply (unfold Lift_def, constrains)
   224 apply (blast dest: zle_imp_zless_or_eq)
   225 done
   226 
   227 lemma bounded: "Lift \<in> Always bounded"
   228 apply (cut_tac moving_up moving_down)
   229 apply (rule AlwaysI, force) 
   230 apply (unfold Lift_def, constrains, auto)
   231 apply (drule not_mem_distinct, assumption, arith)+
   232 done
   233 
   234 
   235 subsection{*Progress*}
   236 
   237 declare moving_def [THEN def_set_simp, simp]
   238 declare stopped_def [THEN def_set_simp, simp]
   239 declare opened_def [THEN def_set_simp, simp]
   240 declare closed_def [THEN def_set_simp, simp]
   241 declare atFloor_def [THEN def_set_simp, simp]
   242 declare Req_def [THEN def_set_simp, simp]
   243 
   244 
   245 (** The HUG'93 paper mistakenly omits the Req n from these! **)
   246 
   247 (** Lift_1 **)
   248 lemma E_thm01: "Lift \<in> (stopped \<inter> atFloor n) LeadsTo (opened \<inter> atFloor n)"
   249 apply (cut_tac stop_floor)
   250 apply (unfold Lift_def, ensures_tac "open_act")
   251 done  (*lem_lift_1_5*)
   252 
   253 
   254 
   255 
   256 lemma E_thm02: "Lift \<in> (Req n \<inter> stopped - atFloor n) LeadsTo  
   257                        (Req n \<inter> opened - atFloor n)"
   258 apply (cut_tac stop_floor)
   259 apply (unfold Lift_def, ensures_tac "open_act")
   260 done  (*lem_lift_1_1*)
   261 
   262 lemma E_thm03: "Lift \<in> (Req n \<inter> opened - atFloor n) LeadsTo  
   263                        (Req n \<inter> closed - (atFloor n - queueing))"
   264 apply (unfold Lift_def, ensures_tac "close_act")
   265 done  (*lem_lift_1_2*)
   266 
   267 lemma E_thm04: "Lift \<in> (Req n \<inter> closed \<inter> (atFloor n - queueing))   
   268                        LeadsTo (opened \<inter> atFloor n)"
   269 apply (unfold Lift_def, ensures_tac "open_act")
   270 done  (*lem_lift_1_7*)
   271 
   272 
   273 (** Lift 2.  Statements of thm05a and thm05b were wrong! **)
   274 
   275 lemmas linorder_leI = linorder_not_less [THEN iffD1]
   276 
   277 lemmas (in Floor) le_MinD = Min_le_n [THEN order_antisym]
   278               and Max_leD = n_le_Max [THEN [2] order_antisym]
   279 
   280 declare (in Floor) le_MinD [dest!]
   281                and linorder_leI [THEN le_MinD, dest!]
   282                and Max_leD [dest!]
   283                and linorder_leI [THEN Max_leD, dest!]
   284 
   285 
   286 (*lem_lift_2_0 
   287   NOT an ensures_tac property, but a mere inclusion
   288   don't know why script lift_2.uni says ENSURES*)
   289 lemma (in Floor) E_thm05c: 
   290     "Lift \<in> (Req n \<inter> closed - (atFloor n - queueing))    
   291              LeadsTo ((closed \<inter> goingup \<inter> Req n)  \<union> 
   292                       (closed \<inter> goingdown \<inter> Req n))"
   293 by (auto intro!: subset_imp_LeadsTo elim!: int_neqE)
   294 
   295 (*lift_2*)
   296 lemma (in Floor) lift_2: "Lift \<in> (Req n \<inter> closed - (atFloor n - queueing))    
   297              LeadsTo (moving \<inter> Req n)"
   298 apply (rule LeadsTo_Trans [OF E_thm05c LeadsTo_Un])
   299 apply (unfold Lift_def) 
   300 apply (ensures_tac [2] "req_down")
   301 apply (ensures_tac "req_up", auto)
   302 done
   303 
   304 
   305 (** Towards lift_4 ***)
   306  
   307 declare split_if_asm [split]
   308 
   309 
   310 (*lem_lift_4_1 *)
   311 lemma (in Floor) E_thm12a:
   312      "0 < N ==>  
   313       Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> 
   314               {s. floor s \<notin> req s} \<inter> {s. up s})    
   315              LeadsTo  
   316                (moving \<inter> Req n \<inter> {s. metric n s < N})"
   317 apply (cut_tac moving_up)
   318 apply (unfold Lift_def, ensures_tac "move_up", safe)
   319 (*this step consolidates two formulae to the goal  metric n s' \<le> metric n s*)
   320 apply (erule linorder_leI [THEN order_antisym, symmetric])
   321 apply (auto simp add: metric_def)
   322 done
   323 
   324 
   325 (*lem_lift_4_3 *)
   326 lemma (in Floor) E_thm12b: "0 < N ==>  
   327       Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> 
   328               {s. floor s \<notin> req s} - {s. up s})    
   329              LeadsTo (moving \<inter> Req n \<inter> {s. metric n s < N})"
   330 apply (cut_tac moving_down)
   331 apply (unfold Lift_def, ensures_tac "move_down", safe)
   332 (*this step consolidates two formulae to the goal  metric n s' \<le> metric n s*)
   333 apply (erule linorder_leI [THEN order_antisym, symmetric])
   334 apply (auto simp add: metric_def)
   335 done
   336 
   337 (*lift_4*)
   338 lemma (in Floor) lift_4:
   339      "0<N ==> Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> 
   340                             {s. floor s \<notin> req s}) LeadsTo      
   341                            (moving \<inter> Req n \<inter> {s. metric n s < N})"
   342 apply (rule LeadsTo_Trans [OF subset_imp_LeadsTo
   343                               LeadsTo_Un [OF E_thm12a E_thm12b]], auto)
   344 done
   345 
   346 
   347 (** towards lift_5 **)
   348 
   349 (*lem_lift_5_3*)
   350 lemma (in Floor) E_thm16a: "0<N    
   351   ==> Lift \<in> (closed \<inter> Req n \<inter> {s. metric n s = N} \<inter> goingup) LeadsTo  
   352              (moving \<inter> Req n \<inter> {s. metric n s < N})"
   353 apply (cut_tac bounded)
   354 apply (unfold Lift_def, ensures_tac "req_up")
   355 apply (auto simp add: metric_def)
   356 done
   357 
   358 
   359 (*lem_lift_5_1 has ~goingup instead of goingdown*)
   360 lemma (in Floor) E_thm16b: "0<N ==>    
   361       Lift \<in> (closed \<inter> Req n \<inter> {s. metric n s = N} \<inter> goingdown) LeadsTo  
   362                    (moving \<inter> Req n \<inter> {s. metric n s < N})"
   363 apply (cut_tac bounded)
   364 apply (unfold Lift_def, ensures_tac "req_down")
   365 apply (auto simp add: metric_def)
   366 done
   367 
   368 
   369 (*lem_lift_5_0 proves an intersection involving ~goingup and goingup,
   370   i.e. the trivial disjunction, leading to an asymmetrical proof.*)
   371 lemma (in Floor) E_thm16c:
   372      "0<N ==> Req n \<inter> {s. metric n s = N} \<subseteq> goingup \<union> goingdown"
   373 by (force simp add: metric_def)
   374 
   375 
   376 (*lift_5*)
   377 lemma (in Floor) lift_5:
   378      "0<N ==> Lift \<in> (closed \<inter> Req n \<inter> {s. metric n s = N}) LeadsTo    
   379                      (moving \<inter> Req n \<inter> {s. metric n s < N})"
   380 apply (rule LeadsTo_Trans [OF subset_imp_LeadsTo
   381                               LeadsTo_Un [OF E_thm16a E_thm16b]])
   382 apply (drule E_thm16c, auto)
   383 done
   384 
   385 
   386 (** towards lift_3 **)
   387 
   388 (*lemma used to prove lem_lift_3_1*)
   389 lemma (in Floor) metric_eq_0D [dest]:
   390      "[| metric n s = 0;  Min \<le> floor s;  floor s \<le> Max |] ==> floor s = n"
   391 by (force simp add: metric_def)
   392 
   393 
   394 (*lem_lift_3_1*)
   395 lemma (in Floor) E_thm11: "Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = 0}) LeadsTo    
   396                        (stopped \<inter> atFloor n)"
   397 apply (cut_tac bounded)
   398 apply (unfold Lift_def, ensures_tac "request_act", auto)
   399 done
   400 
   401 (*lem_lift_3_5*)
   402 lemma (in Floor) E_thm13: 
   403   "Lift \<in> (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s})  
   404   LeadsTo (stopped \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s})"
   405 apply (unfold Lift_def, ensures_tac "request_act")
   406 apply (auto simp add: metric_def)
   407 done
   408 
   409 (*lem_lift_3_6*)
   410 lemma (in Floor) E_thm14: "0 < N ==>  
   411       Lift \<in>  
   412         (stopped \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s})  
   413         LeadsTo (opened \<inter> Req n \<inter> {s. metric n s = N})"
   414 apply (unfold Lift_def, ensures_tac "open_act")
   415 apply (auto simp add: metric_def)
   416 done
   417 
   418 (*lem_lift_3_7*)
   419 lemma (in Floor) E_thm15: "Lift \<in> (opened \<inter> Req n \<inter> {s. metric n s = N})   
   420              LeadsTo (closed \<inter> Req n \<inter> {s. metric n s = N})"
   421 apply (unfold Lift_def, ensures_tac "close_act")
   422 apply (auto simp add: metric_def)
   423 done
   424 
   425 
   426 (** the final steps **)
   427 
   428 lemma (in Floor) lift_3_Req: "0 < N ==>  
   429       Lift \<in>  
   430         (moving \<inter> Req n \<inter> {s. metric n s = N} \<inter> {s. floor s \<in> req s})    
   431         LeadsTo (moving \<inter> Req n \<inter> {s. metric n s < N})"
   432 apply (blast intro!: E_thm13 E_thm14 E_thm15 lift_5 intro: LeadsTo_Trans)
   433 done
   434 
   435 
   436 (*Now we observe that our integer metric is really a natural number*)
   437 lemma (in Floor) Always_nonneg: "Lift \<in> Always {s. 0 \<le> metric n s}"
   438 apply (rule bounded [THEN Always_weaken])
   439 apply (auto simp add: metric_def)
   440 done
   441 
   442 lemmas (in Floor) R_thm11 = Always_LeadsTo_weaken [OF Always_nonneg E_thm11]
   443 
   444 lemma (in Floor) lift_3:
   445      "Lift \<in> (moving \<inter> Req n) LeadsTo (stopped \<inter> atFloor n)"
   446 apply (rule Always_nonneg [THEN integ_0_le_induct])
   447 apply (case_tac "0 < z")
   448 (*If z \<le> 0 then actually z = 0*)
   449 prefer 2 apply (force intro: R_thm11 order_antisym simp add: linorder_not_less)
   450 apply (rule LeadsTo_weaken_R [OF asm_rl Un_upper1])
   451 apply (rule LeadsTo_Trans [OF subset_imp_LeadsTo
   452                               LeadsTo_Un [OF lift_4 lift_3_Req]], auto)
   453 done
   454 
   455 
   456 lemma (in Floor) lift_1: "Lift \<in> (Req n) LeadsTo (opened \<inter> atFloor n)"
   457 apply (rule LeadsTo_Trans)
   458  prefer 2
   459  apply (rule LeadsTo_Un [OF E_thm04 LeadsTo_Un_post])
   460  apply (rule E_thm01 [THEN [2] LeadsTo_Trans_Un])
   461  apply (rule lift_3 [THEN [2] LeadsTo_Trans_Un])
   462  apply (rule lift_2 [THEN [2] LeadsTo_Trans_Un])
   463  apply (rule LeadsTo_Trans_Un [OF E_thm02 E_thm03])
   464 apply (rule open_move [THEN Always_LeadsToI])
   465 apply (rule Always_LeadsToI [OF open_stop subset_imp_LeadsTo], clarify)
   466 (*The case split is not essential but makes the proof much faster.*)
   467 apply (case_tac "open x", auto)
   468 done
   469 
   470 
   471 end