src/HOL/UNITY/Simple/Lift.thy
 author paulson Fri Jan 24 14:06:49 2003 +0100 (2003-01-24) changeset 13785 e2fcd88be55d parent 11868 56db9f3a6b3e child 13806 fd40c9d9076b permissions -rw-r--r--
Partial conversion of UNITY to Isar new-style theories
```     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 <= Max"
```
```    25
```
```    26 constdefs
```
```    27
```
```    28   (** Abbreviations: the "always" part **)
```
```    29
```
```    30   above :: "state set"
```
```    31     "above == {s. EX i. floor s < i & i <= Max & i : req s}"
```
```    32
```
```    33   below :: "state set"
```
```    34     "below == {s. EX i. Min <= i & i < floor s & i : req s}"
```
```    35
```
```    36   queueing :: "state set"
```
```    37     "queueing == above Un below"
```
```    38
```
```    39   goingup :: "state set"
```
```    40     "goingup   == above Int  ({s. up s}  Un -below)"
```
```    41
```
```    42   goingdown :: "state set"
```
```    43     "goingdown == below Int ({s. ~ up s} Un -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 : 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 : 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 : req s
```
```    82 	               & ~(move s & s: 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 : (ready Int 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 : (ready Int 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 ~: 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 ~: 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'). EX n. s' = s (|req := insert n (req s)|)
```
```   118 		        & Min <= n & n <= Max}"
```
```   119
```
```   120
```
```   121   Lift :: "state program"
```
```   122     (*for the moment, we OMIT button_press*)
```
```   123     "Lift == mk_program ({s. floor s = Min & ~ up s & move s & stop s &
```
```   124 		          ~ open s & req s = {}},
```
```   125 			 {request_act, open_act, close_act,
```
```   126 			  req_up, req_down, move_up, move_down},
```
```   127 			 UNIV)"
```
```   128
```
```   129
```
```   130   (** Invariants **)
```
```   131
```
```   132   bounded :: "state set"
```
```   133     "bounded == {s. Min <= floor s & floor s <= Max}"
```
```   134
```
```   135   open_stop :: "state set"
```
```   136     "open_stop == {s. open s --> stop s}"
```
```   137
```
```   138   open_move :: "state set"
```
```   139     "open_move == {s. open s --> move s}"
```
```   140
```
```   141   stop_floor :: "state set"
```
```   142     "stop_floor == {s. stop s & ~ move s --> floor s : req s}"
```
```   143
```
```   144   moving_up :: "state set"
```
```   145     "moving_up == {s. ~ stop s & up s -->
```
```   146                    (EX f. floor s <= f & f <= Max & f : req s)}"
```
```   147
```
```   148   moving_down :: "state set"
```
```   149     "moving_down == {s. ~ stop s & ~ up s -->
```
```   150                      (EX f. Min <= f & f <= floor s & f : req s)}"
```
```   151
```
```   152   metric :: "[int,state] => int"
```
```   153     "metric ==
```
```   154        %n s. if floor s < n then (if up s then n - floor s
```
```   155 			          else (floor s - Min) + (n-Min))
```
```   156              else
```
```   157              if n < floor s then (if up s then (Max - floor s) + (Max-n)
```
```   158 		                  else floor s - n)
```
```   159              else 0"
```
```   160
```
```   161 locale Floor =
```
```   162   fixes n
```
```   163   assumes Min_le_n [iff]: "Min <= n"
```
```   164       and n_le_Max [iff]: "n <= Max"
```
```   165
```
```   166 lemma not_mem_distinct: "[| x ~: A;  y : A |] ==> x ~= y"
```
```   167 by blast
```
```   168
```
```   169
```
```   170 declare Lift_def [THEN def_prg_Init, simp]
```
```   171
```
```   172 declare request_act_def [THEN def_act_simp, simp]
```
```   173 declare open_act_def [THEN def_act_simp, simp]
```
```   174 declare close_act_def [THEN def_act_simp, simp]
```
```   175 declare req_up_def [THEN def_act_simp, simp]
```
```   176 declare req_down_def [THEN def_act_simp, simp]
```
```   177 declare move_up_def [THEN def_act_simp, simp]
```
```   178 declare move_down_def [THEN def_act_simp, simp]
```
```   179 declare button_press_def [THEN def_act_simp, simp]
```
```   180
```
```   181 (*The ALWAYS properties*)
```
```   182 declare above_def [THEN def_set_simp, simp]
```
```   183 declare below_def [THEN def_set_simp, simp]
```
```   184 declare queueing_def [THEN def_set_simp, simp]
```
```   185 declare goingup_def [THEN def_set_simp, simp]
```
```   186 declare goingdown_def [THEN def_set_simp, simp]
```
```   187 declare ready_def [THEN def_set_simp, simp]
```
```   188
```
```   189 (*Basic definitions*)
```
```   190 declare bounded_def [simp]
```
```   191         open_stop_def [simp]
```
```   192         open_move_def [simp]
```
```   193         stop_floor_def [simp]
```
```   194         moving_up_def [simp]
```
```   195         moving_down_def [simp]
```
```   196
```
```   197 lemma open_stop: "Lift : Always open_stop"
```
```   198 apply (rule AlwaysI, force)
```
```   199 apply (unfold Lift_def, constrains)
```
```   200 done
```
```   201
```
```   202 lemma stop_floor: "Lift : Always stop_floor"
```
```   203 apply (rule AlwaysI, force)
```
```   204 apply (unfold Lift_def, constrains)
```
```   205 done
```
```   206
```
```   207 (*This one needs open_stop, which was proved above*)
```
```   208 lemma open_move: "Lift : Always open_move"
```
```   209 apply (cut_tac open_stop)
```
```   210 apply (rule AlwaysI, force)
```
```   211 apply (unfold Lift_def, constrains)
```
```   212 done
```
```   213
```
```   214 lemma moving_up: "Lift : Always moving_up"
```
```   215 apply (rule AlwaysI, force)
```
```   216 apply (unfold Lift_def, constrains)
```
```   217 apply (auto dest: zle_imp_zless_or_eq simp add: add1_zle_eq)
```
```   218 done
```
```   219
```
```   220 lemma moving_down: "Lift : Always moving_down"
```
```   221 apply (rule AlwaysI, force)
```
```   222 apply (unfold Lift_def, constrains)
```
```   223 apply (blast dest: zle_imp_zless_or_eq)
```
```   224 done
```
```   225
```
```   226 lemma bounded: "Lift : Always bounded"
```
```   227 apply (cut_tac moving_up moving_down)
```
```   228 apply (rule AlwaysI, force)
```
```   229 apply (unfold Lift_def, constrains, auto)
```
```   230 apply (drule not_mem_distinct, assumption, arith)+
```
```   231 done
```
```   232
```
```   233
```
```   234 subsection{*Progress*}
```
```   235
```
```   236 declare moving_def [THEN def_set_simp, simp]
```
```   237 declare stopped_def [THEN def_set_simp, simp]
```
```   238 declare opened_def [THEN def_set_simp, simp]
```
```   239 declare closed_def [THEN def_set_simp, simp]
```
```   240 declare atFloor_def [THEN def_set_simp, simp]
```
```   241 declare Req_def [THEN def_set_simp, simp]
```
```   242
```
```   243
```
```   244 (** The HUG'93 paper mistakenly omits the Req n from these! **)
```
```   245
```
```   246 (** Lift_1 **)
```
```   247 lemma E_thm01: "Lift : (stopped Int atFloor n) LeadsTo (opened Int atFloor n)"
```
```   248 apply (cut_tac stop_floor)
```
```   249 apply (unfold Lift_def, ensures_tac "open_act")
```
```   250 done  (*lem_lift_1_5*)
```
```   251
```
```   252 lemma E_thm02: "Lift : (Req n Int stopped - atFloor n) LeadsTo
```
```   253                      (Req n Int opened - atFloor n)"
```
```   254 apply (cut_tac stop_floor)
```
```   255 apply (unfold Lift_def, ensures_tac "open_act")
```
```   256 done  (*lem_lift_1_1*)
```
```   257
```
```   258 lemma E_thm03: "Lift : (Req n Int opened - atFloor n) LeadsTo
```
```   259                      (Req n Int closed - (atFloor n - queueing))"
```
```   260 apply (unfold Lift_def, ensures_tac "close_act")
```
```   261 done  (*lem_lift_1_2*)
```
```   262
```
```   263 lemma E_thm04: "Lift : (Req n Int closed Int (atFloor n - queueing))
```
```   264              LeadsTo (opened Int atFloor n)"
```
```   265 apply (unfold Lift_def, ensures_tac "open_act")
```
```   266 done  (*lem_lift_1_7*)
```
```   267
```
```   268
```
```   269 (** Lift 2.  Statements of thm05a and thm05b were wrong! **)
```
```   270
```
```   271 lemmas linorder_leI = linorder_not_less [THEN iffD1]
```
```   272
```
```   273 lemmas (in Floor) le_MinD = Min_le_n [THEN order_antisym]
```
```   274               and Max_leD = n_le_Max [THEN  order_antisym]
```
```   275
```
```   276 declare (in Floor) le_MinD [dest!]
```
```   277                and linorder_leI [THEN le_MinD, dest!]
```
```   278                and Max_leD [dest!]
```
```   279                and linorder_leI [THEN Max_leD, dest!]
```
```   280
```
```   281
```
```   282 (*lem_lift_2_0
```
```   283   NOT an ensures_tac property, but a mere inclusion
```
```   284   don't know why script lift_2.uni says ENSURES*)
```
```   285 lemma (in Floor) E_thm05c:
```
```   286     "Lift : (Req n Int closed - (atFloor n - queueing))
```
```   287              LeadsTo ((closed Int goingup Int Req n)  Un
```
```   288                       (closed Int goingdown Int Req n))"
```
```   289 by (auto intro!: subset_imp_LeadsTo elim!: int_neqE)
```
```   290
```
```   291 (*lift_2*)
```
```   292 lemma (in Floor) lift_2: "Lift : (Req n Int closed - (atFloor n - queueing))
```
```   293              LeadsTo (moving Int Req n)"
```
```   294 apply (rule LeadsTo_Trans [OF E_thm05c LeadsTo_Un])
```
```   295 apply (unfold Lift_def)
```
```   296 apply (ensures_tac  "req_down")
```
```   297 apply (ensures_tac "req_up", auto)
```
```   298 done
```
```   299
```
```   300
```
```   301 (** Towards lift_4 ***)
```
```   302
```
```   303 declare split_if_asm [split]
```
```   304
```
```   305
```
```   306 (*lem_lift_4_1 *)
```
```   307 lemma (in Floor) E_thm12a:
```
```   308      "0 < N ==>
```
```   309       Lift : (moving Int Req n Int {s. metric n s = N} Int
```
```   310               {s. floor s ~: req s} Int {s. up s})
```
```   311              LeadsTo
```
```   312                (moving Int Req n Int {s. metric n s < N})"
```
```   313 apply (cut_tac moving_up)
```
```   314 apply (unfold Lift_def, ensures_tac "move_up", safe)
```
```   315 (*this step consolidates two formulae to the goal  metric n s' <= metric n s*)
```
```   316 apply (erule linorder_leI [THEN order_antisym, symmetric])
```
```   317 apply (auto simp add: metric_def)
```
```   318 done
```
```   319
```
```   320
```
```   321 (*lem_lift_4_3 *)
```
```   322 lemma (in Floor) E_thm12b: "0 < N ==>
```
```   323       Lift : (moving Int Req n Int {s. metric n s = N} Int
```
```   324               {s. floor s ~: req s} - {s. up s})
```
```   325              LeadsTo (moving Int Req n Int {s. metric n s < N})"
```
```   326 apply (cut_tac moving_down)
```
```   327 apply (unfold Lift_def, ensures_tac "move_down", safe)
```
```   328 (*this step consolidates two formulae to the goal  metric n s' <= metric n s*)
```
```   329 apply (erule linorder_leI [THEN order_antisym, symmetric])
```
```   330 apply (auto simp add: metric_def)
```
```   331 done
```
```   332
```
```   333 (*lift_4*)
```
```   334 lemma (in Floor) lift_4:
```
```   335      "0<N ==> Lift : (moving Int Req n Int {s. metric n s = N} Int
```
```   336                             {s. floor s ~: req s}) LeadsTo
```
```   337                            (moving Int Req n Int {s. metric n s < N})"
```
```   338 apply (rule LeadsTo_Trans [OF subset_imp_LeadsTo
```
```   339                               LeadsTo_Un [OF E_thm12a E_thm12b]], auto)
```
```   340 done
```
```   341
```
```   342
```
```   343 (** towards lift_5 **)
```
```   344
```
```   345 (*lem_lift_5_3*)
```
```   346 lemma (in Floor) E_thm16a: "0<N
```
```   347   ==> Lift : (closed Int Req n Int {s. metric n s = N} Int goingup) LeadsTo
```
```   348              (moving Int Req n Int {s. metric n s < N})"
```
```   349 apply (cut_tac bounded)
```
```   350 apply (unfold Lift_def, ensures_tac "req_up")
```
```   351 apply (auto simp add: metric_def)
```
```   352 done
```
```   353
```
```   354
```
```   355 (*lem_lift_5_1 has ~goingup instead of goingdown*)
```
```   356 lemma (in Floor) E_thm16b: "0<N ==>
```
```   357       Lift : (closed Int Req n Int {s. metric n s = N} Int goingdown) LeadsTo
```
```   358                    (moving Int Req n Int {s. metric n s < N})"
```
```   359 apply (cut_tac bounded)
```
```   360 apply (unfold Lift_def, ensures_tac "req_down")
```
```   361 apply (auto simp add: metric_def)
```
```   362 done
```
```   363
```
```   364
```
```   365 (*lem_lift_5_0 proves an intersection involving ~goingup and goingup,
```
```   366   i.e. the trivial disjunction, leading to an asymmetrical proof.*)
```
```   367 lemma (in Floor) E_thm16c:
```
```   368      "0<N ==> Req n Int {s. metric n s = N} <= goingup Un goingdown"
```
```   369 by (force simp add: metric_def)
```
```   370
```
```   371
```
```   372 (*lift_5*)
```
```   373 lemma (in Floor) lift_5:
```
```   374      "0<N ==> Lift : (closed Int Req n Int {s. metric n s = N}) LeadsTo
```
```   375                      (moving Int Req n Int {s. metric n s < N})"
```
```   376 apply (rule LeadsTo_Trans [OF subset_imp_LeadsTo
```
```   377                               LeadsTo_Un [OF E_thm16a E_thm16b]])
```
```   378 apply (drule E_thm16c, auto)
```
```   379 done
```
```   380
```
```   381
```
```   382 (** towards lift_3 **)
```
```   383
```
```   384 (*lemma used to prove lem_lift_3_1*)
```
```   385 lemma (in Floor) metric_eq_0D [dest]:
```
```   386      "[| metric n s = 0;  Min <= floor s;  floor s <= Max |] ==> floor s = n"
```
```   387 by (force simp add: metric_def)
```
```   388
```
```   389
```
```   390 (*lem_lift_3_1*)
```
```   391 lemma (in Floor) E_thm11: "Lift : (moving Int Req n Int {s. metric n s = 0}) LeadsTo
```
```   392                        (stopped Int atFloor n)"
```
```   393 apply (cut_tac bounded)
```
```   394 apply (unfold Lift_def, ensures_tac "request_act", auto)
```
```   395 done
```
```   396
```
```   397 (*lem_lift_3_5*)
```
```   398 lemma (in Floor) E_thm13:
```
```   399   "Lift : (moving Int Req n Int {s. metric n s = N} Int {s. floor s : req s})
```
```   400   LeadsTo (stopped Int Req n Int {s. metric n s = N} Int {s. floor s : req s})"
```
```   401 apply (unfold Lift_def, ensures_tac "request_act")
```
```   402 apply (auto simp add: metric_def)
```
```   403 done
```
```   404
```
```   405 (*lem_lift_3_6*)
```
```   406 lemma (in Floor) E_thm14: "0 < N ==>
```
```   407       Lift :
```
```   408         (stopped Int Req n Int {s. metric n s = N} Int {s. floor s : req s})
```
```   409         LeadsTo (opened Int Req n Int {s. metric n s = N})"
```
```   410 apply (unfold Lift_def, ensures_tac "open_act")
```
```   411 apply (auto simp add: metric_def)
```
```   412 done
```
```   413
```
```   414 (*lem_lift_3_7*)
```
```   415 lemma (in Floor) E_thm15: "Lift : (opened Int Req n Int {s. metric n s = N})
```
```   416              LeadsTo (closed Int Req n Int {s. metric n s = N})"
```
```   417 apply (unfold Lift_def, ensures_tac "close_act")
```
```   418 apply (auto simp add: metric_def)
```
```   419 done
```
```   420
```
```   421
```
```   422 (** the final steps **)
```
```   423
```
```   424 lemma (in Floor) lift_3_Req: "0 < N ==>
```
```   425       Lift :
```
```   426         (moving Int Req n Int {s. metric n s = N} Int {s. floor s : req s})
```
```   427         LeadsTo (moving Int Req n Int {s. metric n s < N})"
```
```   428 apply (blast intro!: E_thm13 E_thm14 E_thm15 lift_5 intro: LeadsTo_Trans)
```
```   429 done
```
```   430
```
```   431
```
```   432 (*Now we observe that our integer metric is really a natural number*)
```
```   433 lemma (in Floor) Always_nonneg: "Lift : Always {s. 0 <= metric n s}"
```
```   434 apply (rule bounded [THEN Always_weaken])
```
```   435 apply (auto simp add: metric_def)
```
```   436 done
```
```   437
```
```   438 lemmas (in Floor) R_thm11 = Always_LeadsTo_weaken [OF Always_nonneg E_thm11]
```
```   439
```
```   440 lemma (in Floor) lift_3:
```
```   441      "Lift : (moving Int Req n) LeadsTo (stopped Int atFloor n)"
```
```   442 apply (rule Always_nonneg [THEN integ_0_le_induct])
```
```   443 apply (case_tac "0 < z")
```
```   444 (*If z <= 0 then actually z = 0*)
```
```   445 prefer 2 apply (force intro: R_thm11 order_antisym simp add: linorder_not_less)
```
```   446 apply (rule LeadsTo_weaken_R [OF asm_rl Un_upper1])
```
```   447 apply (rule LeadsTo_Trans [OF subset_imp_LeadsTo
```
```   448                               LeadsTo_Un [OF lift_4 lift_3_Req]], auto)
```
```   449 done
```
```   450
```
```   451
```
```   452 lemma (in Floor) lift_1: "Lift : (Req n) LeadsTo (opened Int atFloor n)"
```
```   453 apply (rule LeadsTo_Trans)
```
```   454  prefer 2
```
```   455  apply (rule LeadsTo_Un [OF E_thm04 LeadsTo_Un_post])
```
```   456  apply (rule E_thm01 [THEN  LeadsTo_Trans_Un])
```
```   457  apply (rule lift_3 [THEN  LeadsTo_Trans_Un])
```
```   458  apply (rule lift_2 [THEN  LeadsTo_Trans_Un])
```
```   459  apply (rule LeadsTo_Trans_Un [OF E_thm02 E_thm03])
```
```   460 apply (rule open_move [THEN Always_LeadsToI])
```
```   461 apply (rule Always_LeadsToI [OF open_stop subset_imp_LeadsTo], clarify)
```
```   462 (*The case split is not essential but makes the proof much faster.*)
```
```   463 apply (case_tac "open x", auto)
```
```   464 done
```
```   465
```
```   466
```
```   467 end
```