src/HOL/TLA/TLA.thy
author wenzelm
Fri May 13 23:58:40 2011 +0200 (2011-05-13)
changeset 42795 66fcc9882784
parent 42793 88bee9f6eec7
child 42803 7ed59879b1b6
permissions -rw-r--r--
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm@35108
     1
(*  Title:      HOL/TLA/TLA.thy
wenzelm@35108
     2
    Author:     Stephan Merz
wenzelm@35108
     3
    Copyright:  1998 University of Munich
wenzelm@21624
     4
*)
wenzelm@3807
     5
wenzelm@21624
     6
header {* The temporal level of TLA *}
wenzelm@3807
     7
wenzelm@17309
     8
theory TLA
wenzelm@17309
     9
imports Init
wenzelm@17309
    10
begin
wenzelm@3807
    11
wenzelm@3807
    12
consts
wenzelm@6255
    13
  (** abstract syntax **)
wenzelm@17309
    14
  Box        :: "('w::world) form => temporal"
wenzelm@17309
    15
  Dmd        :: "('w::world) form => temporal"
wenzelm@17309
    16
  leadsto    :: "['w::world form, 'v::world form] => temporal"
wenzelm@17309
    17
  Stable     :: "stpred => temporal"
wenzelm@17309
    18
  WF         :: "[action, 'a stfun] => temporal"
wenzelm@17309
    19
  SF         :: "[action, 'a stfun] => temporal"
wenzelm@3807
    20
wenzelm@3807
    21
  (* Quantification over (flexible) state variables *)
wenzelm@17309
    22
  EEx        :: "('a stfun => temporal) => temporal"       (binder "Eex " 10)
wenzelm@17309
    23
  AAll       :: "('a stfun => temporal) => temporal"       (binder "Aall " 10)
wenzelm@6255
    24
wenzelm@6255
    25
  (** concrete syntax **)
wenzelm@6255
    26
syntax
wenzelm@17309
    27
  "_Box"     :: "lift => lift"                        ("([]_)" [40] 40)
wenzelm@17309
    28
  "_Dmd"     :: "lift => lift"                        ("(<>_)" [40] 40)
wenzelm@17309
    29
  "_leadsto" :: "[lift,lift] => lift"                 ("(_ ~> _)" [23,22] 22)
wenzelm@17309
    30
  "_stable"  :: "lift => lift"                        ("(stable/ _)")
wenzelm@17309
    31
  "_WF"      :: "[lift,lift] => lift"                 ("(WF'(_')'_(_))" [0,60] 55)
wenzelm@17309
    32
  "_SF"      :: "[lift,lift] => lift"                 ("(SF'(_')'_(_))" [0,60] 55)
wenzelm@6255
    33
wenzelm@17309
    34
  "_EEx"     :: "[idts, lift] => lift"                ("(3EEX _./ _)" [0,10] 10)
wenzelm@17309
    35
  "_AAll"    :: "[idts, lift] => lift"                ("(3AALL _./ _)" [0,10] 10)
wenzelm@3807
    36
wenzelm@3807
    37
translations
wenzelm@35068
    38
  "_Box"      ==   "CONST Box"
wenzelm@35068
    39
  "_Dmd"      ==   "CONST Dmd"
wenzelm@35068
    40
  "_leadsto"  ==   "CONST leadsto"
wenzelm@35068
    41
  "_stable"   ==   "CONST Stable"
wenzelm@35068
    42
  "_WF"       ==   "CONST WF"
wenzelm@35068
    43
  "_SF"       ==   "CONST SF"
wenzelm@6255
    44
  "_EEx v A"  ==   "Eex v. A"
wenzelm@6255
    45
  "_AAll v A" ==   "Aall v. A"
wenzelm@6255
    46
wenzelm@6255
    47
  "sigma |= []F"         <= "_Box F sigma"
wenzelm@6255
    48
  "sigma |= <>F"         <= "_Dmd F sigma"
wenzelm@6255
    49
  "sigma |= F ~> G"      <= "_leadsto F G sigma"
wenzelm@6255
    50
  "sigma |= stable P"    <= "_stable P sigma"
wenzelm@6255
    51
  "sigma |= WF(A)_v"     <= "_WF A v sigma"
wenzelm@6255
    52
  "sigma |= SF(A)_v"     <= "_SF A v sigma"
wenzelm@6255
    53
  "sigma |= EEX x. F"    <= "_EEx x F sigma"
wenzelm@6255
    54
  "sigma |= AALL x. F"    <= "_AAll x F sigma"
wenzelm@3807
    55
wenzelm@12114
    56
syntax (xsymbols)
wenzelm@17309
    57
  "_Box"     :: "lift => lift"                        ("(\<box>_)" [40] 40)
wenzelm@17309
    58
  "_Dmd"     :: "lift => lift"                        ("(\<diamond>_)" [40] 40)
wenzelm@17309
    59
  "_leadsto" :: "[lift,lift] => lift"                 ("(_ \<leadsto> _)" [23,22] 22)
wenzelm@17309
    60
  "_EEx"     :: "[idts, lift] => lift"                ("(3\<exists>\<exists> _./ _)" [0,10] 10)
wenzelm@17309
    61
  "_AAll"    :: "[idts, lift] => lift"                ("(3\<forall>\<forall> _./ _)" [0,10] 10)
wenzelm@3808
    62
kleing@14565
    63
syntax (HTML output)
wenzelm@17309
    64
  "_EEx"     :: "[idts, lift] => lift"                ("(3\<exists>\<exists> _./ _)" [0,10] 10)
wenzelm@17309
    65
  "_AAll"    :: "[idts, lift] => lift"                ("(3\<forall>\<forall> _./ _)" [0,10] 10)
kleing@14565
    66
wenzelm@17309
    67
axioms
wenzelm@6255
    68
  (* Definitions of derived operators *)
wenzelm@17309
    69
  dmd_def:      "TEMP <>F  ==  TEMP ~[]~F"
wenzelm@17309
    70
  boxInit:      "TEMP []F  ==  TEMP []Init F"
wenzelm@17309
    71
  leadsto_def:  "TEMP F ~> G  ==  TEMP [](Init F --> <>G)"
wenzelm@17309
    72
  stable_def:   "TEMP stable P  ==  TEMP []($P --> P$)"
wenzelm@17309
    73
  WF_def:       "TEMP WF(A)_v  ==  TEMP <>[] Enabled(<A>_v) --> []<><A>_v"
wenzelm@17309
    74
  SF_def:       "TEMP SF(A)_v  ==  TEMP []<> Enabled(<A>_v) --> []<><A>_v"
wenzelm@17309
    75
  aall_def:     "TEMP (AALL x. F x)  ==  TEMP ~ (EEX x. ~ F x)"
wenzelm@3807
    76
wenzelm@6255
    77
(* Base axioms for raw TLA. *)
wenzelm@17309
    78
  normalT:    "|- [](F --> G) --> ([]F --> []G)"    (* polymorphic *)
wenzelm@17309
    79
  reflT:      "|- []F --> F"         (* F::temporal *)
wenzelm@17309
    80
  transT:     "|- []F --> [][]F"     (* polymorphic *)
wenzelm@17309
    81
  linT:       "|- <>F & <>G --> (<>(F & <>G)) | (<>(G & <>F))"
wenzelm@17309
    82
  discT:      "|- [](F --> <>(~F & <>F)) --> (F --> []<>F)"
wenzelm@17309
    83
  primeI:     "|- []P --> Init P`"
wenzelm@17309
    84
  primeE:     "|- [](Init P --> []F) --> Init P` --> (F --> []F)"
wenzelm@17309
    85
  indT:       "|- [](Init P & ~[]F --> Init P` & F) --> Init P --> []F"
wenzelm@17309
    86
  allT:       "|- (ALL x. [](F x)) = ([](ALL x. F x))"
wenzelm@3807
    87
wenzelm@17309
    88
  necT:       "|- F ==> |- []F"      (* polymorphic *)
wenzelm@3807
    89
wenzelm@3807
    90
(* Flexible quantification: refinement mappings, history variables *)
wenzelm@17309
    91
  eexI:       "|- F x --> (EEX x. F x)"
wenzelm@17309
    92
  eexE:       "[| sigma |= (EEX x. F x); basevars vs;
wenzelm@17309
    93
                 (!!x. [| basevars (x, vs); sigma |= F x |] ==> (G sigma)::bool)
wenzelm@17309
    94
              |] ==> G sigma"
wenzelm@17309
    95
  history:    "|- EEX h. Init(h = ha) & [](!x. $h = #x --> h` = hb x)"
wenzelm@17309
    96
wenzelm@21624
    97
wenzelm@21624
    98
(* Specialize intensional introduction/elimination rules for temporal formulas *)
wenzelm@21624
    99
wenzelm@21624
   100
lemma tempI: "(!!sigma. sigma |= (F::temporal)) ==> |- F"
wenzelm@21624
   101
  apply (rule intI)
wenzelm@21624
   102
  apply (erule meta_spec)
wenzelm@21624
   103
  done
wenzelm@21624
   104
wenzelm@21624
   105
lemma tempD: "|- (F::temporal) ==> sigma |= F"
wenzelm@21624
   106
  by (erule intD)
wenzelm@21624
   107
wenzelm@21624
   108
wenzelm@21624
   109
(* ======== Functions to "unlift" temporal theorems ====== *)
wenzelm@21624
   110
wenzelm@21624
   111
ML {*
wenzelm@21624
   112
(* The following functions are specialized versions of the corresponding
wenzelm@21624
   113
   functions defined in theory Intensional in that they introduce a
wenzelm@21624
   114
   "world" parameter of type "behavior".
wenzelm@21624
   115
*)
wenzelm@21624
   116
fun temp_unlift th =
wenzelm@26305
   117
  (rewrite_rule @{thms action_rews} (th RS @{thm tempD})) handle THM _ => action_unlift th;
wenzelm@21624
   118
wenzelm@21624
   119
(* Turn  |- F = G  into meta-level rewrite rule  F == G *)
wenzelm@21624
   120
val temp_rewrite = int_rewrite
wenzelm@21624
   121
wenzelm@21624
   122
fun temp_use th =
wenzelm@21624
   123
  case (concl_of th) of
wenzelm@26305
   124
    Const _ $ (Const (@{const_name Intensional.Valid}, _) $ _) =>
wenzelm@21624
   125
            ((flatten (temp_unlift th)) handle THM _ => th)
wenzelm@21624
   126
  | _ => th;
wenzelm@21624
   127
wenzelm@21624
   128
fun try_rewrite th = temp_rewrite th handle THM _ => temp_use th;
wenzelm@21624
   129
*}
wenzelm@21624
   130
wenzelm@30528
   131
attribute_setup temp_unlift = {* Scan.succeed (Thm.rule_attribute (K temp_unlift)) *} ""
wenzelm@30528
   132
attribute_setup temp_rewrite = {* Scan.succeed (Thm.rule_attribute (K temp_rewrite)) *} ""
wenzelm@30528
   133
attribute_setup temp_use = {* Scan.succeed (Thm.rule_attribute (K temp_use)) *} ""
wenzelm@30528
   134
attribute_setup try_rewrite = {* Scan.succeed (Thm.rule_attribute (K try_rewrite)) *} ""
wenzelm@30528
   135
wenzelm@21624
   136
wenzelm@21624
   137
(* Update classical reasoner---will be updated once more below! *)
wenzelm@21624
   138
wenzelm@21624
   139
declare tempI [intro!]
wenzelm@21624
   140
declare tempD [dest]
wenzelm@21624
   141
wenzelm@21624
   142
(* Modify the functions that add rules to simpsets, classical sets,
wenzelm@21624
   143
   and clasimpsets in order to accept "lifted" theorems
wenzelm@21624
   144
*)
wenzelm@21624
   145
wenzelm@21624
   146
(* ------------------------------------------------------------------------- *)
wenzelm@21624
   147
(***           "Simple temporal logic": only [] and <>                     ***)
wenzelm@21624
   148
(* ------------------------------------------------------------------------- *)
wenzelm@21624
   149
section "Simple temporal logic"
wenzelm@21624
   150
wenzelm@21624
   151
(* []~F == []~Init F *)
wenzelm@21624
   152
lemmas boxNotInit = boxInit [of "LIFT ~F", unfolded Init_simps, standard]
wenzelm@21624
   153
wenzelm@21624
   154
lemma dmdInit: "TEMP <>F == TEMP <> Init F"
wenzelm@21624
   155
  apply (unfold dmd_def)
wenzelm@21624
   156
  apply (unfold boxInit [of "LIFT ~F"])
wenzelm@21624
   157
  apply (simp (no_asm) add: Init_simps)
wenzelm@21624
   158
  done
wenzelm@21624
   159
wenzelm@21624
   160
lemmas dmdNotInit = dmdInit [of "LIFT ~F", unfolded Init_simps, standard]
wenzelm@21624
   161
wenzelm@21624
   162
(* boxInit and dmdInit cannot be used as rewrites, because they loop.
wenzelm@21624
   163
   Non-looping instances for state predicates and actions are occasionally useful.
wenzelm@21624
   164
*)
wenzelm@21624
   165
lemmas boxInit_stp = boxInit [where 'a = state, standard]
wenzelm@21624
   166
lemmas boxInit_act = boxInit [where 'a = "state * state", standard]
wenzelm@21624
   167
lemmas dmdInit_stp = dmdInit [where 'a = state, standard]
wenzelm@21624
   168
lemmas dmdInit_act = dmdInit [where 'a = "state * state", standard]
wenzelm@21624
   169
wenzelm@21624
   170
(* The symmetric equations can be used to get rid of Init *)
wenzelm@21624
   171
lemmas boxInitD = boxInit [symmetric]
wenzelm@21624
   172
lemmas dmdInitD = dmdInit [symmetric]
wenzelm@21624
   173
lemmas boxNotInitD = boxNotInit [symmetric]
wenzelm@21624
   174
lemmas dmdNotInitD = dmdNotInit [symmetric]
wenzelm@21624
   175
wenzelm@21624
   176
lemmas Init_simps = Init_simps boxInitD dmdInitD boxNotInitD dmdNotInitD
wenzelm@21624
   177
wenzelm@21624
   178
(* ------------------------ STL2 ------------------------------------------- *)
wenzelm@21624
   179
lemmas STL2 = reflT
wenzelm@21624
   180
wenzelm@21624
   181
(* The "polymorphic" (generic) variant *)
wenzelm@21624
   182
lemma STL2_gen: "|- []F --> Init F"
wenzelm@21624
   183
  apply (unfold boxInit [of F])
wenzelm@21624
   184
  apply (rule STL2)
wenzelm@21624
   185
  done
wenzelm@21624
   186
wenzelm@21624
   187
(* see also STL2_pr below: "|- []P --> Init P & Init (P`)" *)
wenzelm@21624
   188
wenzelm@21624
   189
wenzelm@21624
   190
(* Dual versions for <> *)
wenzelm@21624
   191
lemma InitDmd: "|- F --> <> F"
wenzelm@21624
   192
  apply (unfold dmd_def)
wenzelm@21624
   193
  apply (auto dest!: STL2 [temp_use])
wenzelm@21624
   194
  done
wenzelm@21624
   195
wenzelm@21624
   196
lemma InitDmd_gen: "|- Init F --> <>F"
wenzelm@21624
   197
  apply clarsimp
wenzelm@21624
   198
  apply (drule InitDmd [temp_use])
wenzelm@21624
   199
  apply (simp add: dmdInitD)
wenzelm@21624
   200
  done
wenzelm@21624
   201
wenzelm@21624
   202
wenzelm@21624
   203
(* ------------------------ STL3 ------------------------------------------- *)
wenzelm@21624
   204
lemma STL3: "|- ([][]F) = ([]F)"
wenzelm@21624
   205
  by (auto elim: transT [temp_use] STL2 [temp_use])
wenzelm@21624
   206
wenzelm@21624
   207
(* corresponding elimination rule introduces double boxes:
wenzelm@21624
   208
   [| (sigma |= []F); (sigma |= [][]F) ==> PROP W |] ==> PROP W
wenzelm@21624
   209
*)
wenzelm@21624
   210
lemmas dup_boxE = STL3 [temp_unlift, THEN iffD2, elim_format]
wenzelm@21624
   211
lemmas dup_boxD = STL3 [temp_unlift, THEN iffD1, standard]
wenzelm@21624
   212
wenzelm@21624
   213
(* dual versions for <> *)
wenzelm@21624
   214
lemma DmdDmd: "|- (<><>F) = (<>F)"
wenzelm@21624
   215
  by (auto simp add: dmd_def [try_rewrite] STL3 [try_rewrite])
wenzelm@21624
   216
wenzelm@21624
   217
lemmas dup_dmdE = DmdDmd [temp_unlift, THEN iffD2, elim_format]
wenzelm@21624
   218
lemmas dup_dmdD = DmdDmd [temp_unlift, THEN iffD1, standard]
wenzelm@21624
   219
wenzelm@21624
   220
wenzelm@21624
   221
(* ------------------------ STL4 ------------------------------------------- *)
wenzelm@21624
   222
lemma STL4:
wenzelm@21624
   223
  assumes "|- F --> G"
wenzelm@21624
   224
  shows "|- []F --> []G"
wenzelm@21624
   225
  apply clarsimp
wenzelm@21624
   226
  apply (rule normalT [temp_use])
wenzelm@21624
   227
   apply (rule assms [THEN necT, temp_use])
wenzelm@21624
   228
  apply assumption
wenzelm@21624
   229
  done
wenzelm@21624
   230
wenzelm@21624
   231
(* Unlifted version as an elimination rule *)
wenzelm@21624
   232
lemma STL4E: "[| sigma |= []F; |- F --> G |] ==> sigma |= []G"
wenzelm@21624
   233
  by (erule (1) STL4 [temp_use])
wenzelm@21624
   234
wenzelm@21624
   235
lemma STL4_gen: "|- Init F --> Init G ==> |- []F --> []G"
wenzelm@21624
   236
  apply (drule STL4)
wenzelm@21624
   237
  apply (simp add: boxInitD)
wenzelm@21624
   238
  done
wenzelm@21624
   239
wenzelm@21624
   240
lemma STL4E_gen: "[| sigma |= []F; |- Init F --> Init G |] ==> sigma |= []G"
wenzelm@21624
   241
  by (erule (1) STL4_gen [temp_use])
wenzelm@21624
   242
wenzelm@21624
   243
(* see also STL4Edup below, which allows an auxiliary boxed formula:
wenzelm@21624
   244
       []A /\ F => G
wenzelm@21624
   245
     -----------------
wenzelm@21624
   246
     []A /\ []F => []G
wenzelm@21624
   247
*)
wenzelm@21624
   248
wenzelm@21624
   249
(* The dual versions for <> *)
wenzelm@21624
   250
lemma DmdImpl:
wenzelm@21624
   251
  assumes prem: "|- F --> G"
wenzelm@21624
   252
  shows "|- <>F --> <>G"
wenzelm@21624
   253
  apply (unfold dmd_def)
wenzelm@21624
   254
  apply (fastsimp intro!: prem [temp_use] elim!: STL4E [temp_use])
wenzelm@21624
   255
  done
wenzelm@21624
   256
wenzelm@21624
   257
lemma DmdImplE: "[| sigma |= <>F; |- F --> G |] ==> sigma |= <>G"
wenzelm@21624
   258
  by (erule (1) DmdImpl [temp_use])
wenzelm@21624
   259
wenzelm@21624
   260
(* ------------------------ STL5 ------------------------------------------- *)
wenzelm@21624
   261
lemma STL5: "|- ([]F & []G) = ([](F & G))"
wenzelm@21624
   262
  apply auto
wenzelm@21624
   263
  apply (subgoal_tac "sigma |= [] (G --> (F & G))")
wenzelm@21624
   264
     apply (erule normalT [temp_use])
wenzelm@21624
   265
     apply (fastsimp elim!: STL4E [temp_use])+
wenzelm@21624
   266
  done
wenzelm@21624
   267
wenzelm@21624
   268
(* rewrite rule to split conjunctions under boxes *)
wenzelm@21624
   269
lemmas split_box_conj = STL5 [temp_unlift, symmetric, standard]
wenzelm@21624
   270
wenzelm@21624
   271
wenzelm@21624
   272
(* the corresponding elimination rule allows to combine boxes in the hypotheses
wenzelm@21624
   273
   (NB: F and G must have the same type, i.e., both actions or temporals.)
wenzelm@21624
   274
   Use "addSE2" etc. if you want to add this to a claset, otherwise it will loop!
wenzelm@21624
   275
*)
wenzelm@21624
   276
lemma box_conjE:
wenzelm@21624
   277
  assumes "sigma |= []F"
wenzelm@21624
   278
     and "sigma |= []G"
wenzelm@21624
   279
  and "sigma |= [](F&G) ==> PROP R"
wenzelm@21624
   280
  shows "PROP R"
wenzelm@21624
   281
  by (rule assms STL5 [temp_unlift, THEN iffD1] conjI)+
wenzelm@21624
   282
wenzelm@21624
   283
(* Instances of box_conjE for state predicates, actions, and temporals
wenzelm@21624
   284
   in case the general rule is "too polymorphic".
wenzelm@21624
   285
*)
wenzelm@21624
   286
lemmas box_conjE_temp = box_conjE [where 'a = behavior, standard]
wenzelm@21624
   287
lemmas box_conjE_stp = box_conjE [where 'a = state, standard]
wenzelm@21624
   288
lemmas box_conjE_act = box_conjE [where 'a = "state * state", standard]
wenzelm@21624
   289
wenzelm@21624
   290
(* Define a tactic that tries to merge all boxes in an antecedent. The definition is
wenzelm@21624
   291
   a bit kludgy in order to simulate "double elim-resolution".
wenzelm@21624
   292
*)
wenzelm@21624
   293
wenzelm@21624
   294
lemma box_thin: "[| sigma |= []F; PROP W |] ==> PROP W" .
wenzelm@21624
   295
wenzelm@21624
   296
ML {*
wenzelm@21624
   297
fun merge_box_tac i =
wenzelm@26305
   298
   REPEAT_DETERM (EVERY [etac @{thm box_conjE} i, atac i, etac @{thm box_thin} i])
wenzelm@21624
   299
wenzelm@27208
   300
fun merge_temp_box_tac ctxt i =
wenzelm@26305
   301
   REPEAT_DETERM (EVERY [etac @{thm box_conjE_temp} i, atac i,
wenzelm@27239
   302
                         eres_inst_tac ctxt [(("'a", 0), "behavior")] @{thm box_thin} i])
wenzelm@21624
   303
wenzelm@27208
   304
fun merge_stp_box_tac ctxt i =
wenzelm@26305
   305
   REPEAT_DETERM (EVERY [etac @{thm box_conjE_stp} i, atac i,
wenzelm@27239
   306
                         eres_inst_tac ctxt [(("'a", 0), "state")] @{thm box_thin} i])
wenzelm@21624
   307
wenzelm@27208
   308
fun merge_act_box_tac ctxt i =
wenzelm@26305
   309
   REPEAT_DETERM (EVERY [etac @{thm box_conjE_act} i, atac i,
wenzelm@27239
   310
                         eres_inst_tac ctxt [(("'a", 0), "state * state")] @{thm box_thin} i])
wenzelm@21624
   311
*}
wenzelm@21624
   312
wenzelm@42787
   313
method_setup merge_box = {* Scan.succeed (K (SIMPLE_METHOD' merge_box_tac)) *} ""
wenzelm@42787
   314
method_setup merge_temp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_temp_box_tac) *} ""
wenzelm@42787
   315
method_setup merge_stp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_stp_box_tac) *} ""
wenzelm@42787
   316
method_setup merge_act_box = {* Scan.succeed (SIMPLE_METHOD' o merge_act_box_tac) *} ""
wenzelm@42787
   317
wenzelm@21624
   318
(* rewrite rule to push universal quantification through box:
wenzelm@21624
   319
      (sigma |= [](! x. F x)) = (! x. (sigma |= []F x))
wenzelm@21624
   320
*)
wenzelm@21624
   321
lemmas all_box = allT [temp_unlift, symmetric, standard]
wenzelm@21624
   322
wenzelm@21624
   323
lemma DmdOr: "|- (<>(F | G)) = (<>F | <>G)"
wenzelm@21624
   324
  apply (auto simp add: dmd_def split_box_conj [try_rewrite])
wenzelm@42787
   325
  apply (erule contrapos_np, merge_box, fastsimp elim!: STL4E [temp_use])+
wenzelm@21624
   326
  done
wenzelm@21624
   327
wenzelm@21624
   328
lemma exT: "|- (EX x. <>(F x)) = (<>(EX x. F x))"
wenzelm@21624
   329
  by (auto simp: dmd_def Not_Rex [try_rewrite] all_box [try_rewrite])
wenzelm@21624
   330
wenzelm@21624
   331
lemmas ex_dmd = exT [temp_unlift, symmetric, standard]
wenzelm@21624
   332
wenzelm@21624
   333
lemma STL4Edup: "!!sigma. [| sigma |= []A; sigma |= []F; |- F & []A --> G |] ==> sigma |= []G"
wenzelm@21624
   334
  apply (erule dup_boxE)
wenzelm@42787
   335
  apply merge_box
wenzelm@21624
   336
  apply (erule STL4E)
wenzelm@21624
   337
  apply assumption
wenzelm@21624
   338
  done
wenzelm@21624
   339
wenzelm@21624
   340
lemma DmdImpl2: 
wenzelm@21624
   341
    "!!sigma. [| sigma |= <>F; sigma |= [](F --> G) |] ==> sigma |= <>G"
wenzelm@21624
   342
  apply (unfold dmd_def)
wenzelm@21624
   343
  apply auto
wenzelm@21624
   344
  apply (erule notE)
wenzelm@42787
   345
  apply merge_box
wenzelm@21624
   346
  apply (fastsimp elim!: STL4E [temp_use])
wenzelm@21624
   347
  done
wenzelm@21624
   348
wenzelm@21624
   349
lemma InfImpl:
wenzelm@21624
   350
  assumes 1: "sigma |= []<>F"
wenzelm@21624
   351
    and 2: "sigma |= []G"
wenzelm@21624
   352
    and 3: "|- F & G --> H"
wenzelm@21624
   353
  shows "sigma |= []<>H"
wenzelm@21624
   354
  apply (insert 1 2)
wenzelm@21624
   355
  apply (erule_tac F = G in dup_boxE)
wenzelm@42787
   356
  apply merge_box
wenzelm@21624
   357
  apply (fastsimp elim!: STL4E [temp_use] DmdImpl2 [temp_use] intro!: 3 [temp_use])
wenzelm@21624
   358
  done
wenzelm@21624
   359
wenzelm@21624
   360
(* ------------------------ STL6 ------------------------------------------- *)
wenzelm@21624
   361
(* Used in the proof of STL6, but useful in itself. *)
wenzelm@21624
   362
lemma BoxDmd: "|- []F & <>G --> <>([]F & G)"
wenzelm@21624
   363
  apply (unfold dmd_def)
wenzelm@21624
   364
  apply clarsimp
wenzelm@21624
   365
  apply (erule dup_boxE)
wenzelm@42787
   366
  apply merge_box
wenzelm@21624
   367
  apply (erule contrapos_np)
wenzelm@21624
   368
  apply (fastsimp elim!: STL4E [temp_use])
wenzelm@21624
   369
  done
wenzelm@21624
   370
wenzelm@21624
   371
(* weaker than BoxDmd, but more polymorphic (and often just right) *)
wenzelm@21624
   372
lemma BoxDmd_simple: "|- []F & <>G --> <>(F & G)"
wenzelm@21624
   373
  apply (unfold dmd_def)
wenzelm@21624
   374
  apply clarsimp
wenzelm@42787
   375
  apply merge_box
wenzelm@21624
   376
  apply (fastsimp elim!: notE STL4E [temp_use])
wenzelm@21624
   377
  done
wenzelm@21624
   378
wenzelm@21624
   379
lemma BoxDmd2_simple: "|- []F & <>G --> <>(G & F)"
wenzelm@21624
   380
  apply (unfold dmd_def)
wenzelm@21624
   381
  apply clarsimp
wenzelm@42787
   382
  apply merge_box
wenzelm@21624
   383
  apply (fastsimp elim!: notE STL4E [temp_use])
wenzelm@21624
   384
  done
wenzelm@21624
   385
wenzelm@21624
   386
lemma DmdImpldup:
wenzelm@21624
   387
  assumes 1: "sigma |= []A"
wenzelm@21624
   388
    and 2: "sigma |= <>F"
wenzelm@21624
   389
    and 3: "|- []A & F --> G"
wenzelm@21624
   390
  shows "sigma |= <>G"
wenzelm@21624
   391
  apply (rule 2 [THEN 1 [THEN BoxDmd [temp_use]], THEN DmdImplE])
wenzelm@21624
   392
  apply (rule 3)
wenzelm@21624
   393
  done
wenzelm@21624
   394
wenzelm@21624
   395
lemma STL6: "|- <>[]F & <>[]G --> <>[](F & G)"
wenzelm@21624
   396
  apply (auto simp: STL5 [temp_rewrite, symmetric])
wenzelm@21624
   397
  apply (drule linT [temp_use])
wenzelm@21624
   398
   apply assumption
wenzelm@21624
   399
  apply (erule thin_rl)
wenzelm@21624
   400
  apply (rule DmdDmd [temp_unlift, THEN iffD1])
wenzelm@21624
   401
  apply (erule disjE)
wenzelm@21624
   402
   apply (erule DmdImplE)
wenzelm@21624
   403
   apply (rule BoxDmd)
wenzelm@21624
   404
  apply (erule DmdImplE)
wenzelm@21624
   405
  apply auto
wenzelm@21624
   406
  apply (drule BoxDmd [temp_use])
wenzelm@21624
   407
   apply assumption
wenzelm@21624
   408
  apply (erule thin_rl)
wenzelm@21624
   409
  apply (fastsimp elim!: DmdImplE [temp_use])
wenzelm@21624
   410
  done
wenzelm@21624
   411
wenzelm@21624
   412
wenzelm@21624
   413
(* ------------------------ True / False ----------------------------------------- *)
wenzelm@21624
   414
section "Simplification of constants"
wenzelm@21624
   415
wenzelm@21624
   416
lemma BoxConst: "|- ([]#P) = #P"
wenzelm@21624
   417
  apply (rule tempI)
wenzelm@21624
   418
  apply (cases P)
wenzelm@21624
   419
   apply (auto intro!: necT [temp_use] dest: STL2_gen [temp_use] simp: Init_simps)
wenzelm@21624
   420
  done
wenzelm@21624
   421
wenzelm@21624
   422
lemma DmdConst: "|- (<>#P) = #P"
wenzelm@21624
   423
  apply (unfold dmd_def)
wenzelm@21624
   424
  apply (cases P)
wenzelm@21624
   425
  apply (simp_all add: BoxConst [try_rewrite])
wenzelm@21624
   426
  done
wenzelm@21624
   427
wenzelm@21624
   428
lemmas temp_simps [temp_rewrite, simp] = BoxConst DmdConst
wenzelm@21624
   429
wenzelm@21624
   430
wenzelm@21624
   431
(* ------------------------ Further rewrites ----------------------------------------- *)
wenzelm@21624
   432
section "Further rewrites"
wenzelm@21624
   433
wenzelm@21624
   434
lemma NotBox: "|- (~[]F) = (<>~F)"
wenzelm@21624
   435
  by (simp add: dmd_def)
wenzelm@21624
   436
wenzelm@21624
   437
lemma NotDmd: "|- (~<>F) = ([]~F)"
wenzelm@21624
   438
  by (simp add: dmd_def)
wenzelm@21624
   439
wenzelm@21624
   440
(* These are not declared by default, because they could be harmful,
wenzelm@21624
   441
   e.g. []F & ~[]F becomes []F & <>~F !! *)
wenzelm@26305
   442
lemmas more_temp_simps1 =
wenzelm@21624
   443
  STL3 [temp_rewrite] DmdDmd [temp_rewrite] NotBox [temp_rewrite] NotDmd [temp_rewrite]
wenzelm@21624
   444
  NotBox [temp_unlift, THEN eq_reflection]
wenzelm@21624
   445
  NotDmd [temp_unlift, THEN eq_reflection]
wenzelm@21624
   446
wenzelm@21624
   447
lemma BoxDmdBox: "|- ([]<>[]F) = (<>[]F)"
wenzelm@21624
   448
  apply (auto dest!: STL2 [temp_use])
wenzelm@21624
   449
  apply (rule ccontr)
wenzelm@21624
   450
  apply (subgoal_tac "sigma |= <>[][]F & <>[]~[]F")
wenzelm@21624
   451
   apply (erule thin_rl)
wenzelm@21624
   452
   apply auto
wenzelm@21624
   453
    apply (drule STL6 [temp_use])
wenzelm@21624
   454
     apply assumption
wenzelm@21624
   455
    apply simp
wenzelm@26305
   456
   apply (simp_all add: more_temp_simps1)
wenzelm@21624
   457
  done
wenzelm@21624
   458
wenzelm@21624
   459
lemma DmdBoxDmd: "|- (<>[]<>F) = ([]<>F)"
wenzelm@21624
   460
  apply (unfold dmd_def)
wenzelm@21624
   461
  apply (auto simp: BoxDmdBox [unfolded dmd_def, try_rewrite])
wenzelm@21624
   462
  done
wenzelm@21624
   463
wenzelm@26305
   464
lemmas more_temp_simps2 = more_temp_simps1 BoxDmdBox [temp_rewrite] DmdBoxDmd [temp_rewrite]
wenzelm@21624
   465
wenzelm@21624
   466
wenzelm@21624
   467
(* ------------------------ Miscellaneous ----------------------------------- *)
wenzelm@21624
   468
wenzelm@21624
   469
lemma BoxOr: "!!sigma. [| sigma |= []F | []G |] ==> sigma |= [](F | G)"
wenzelm@21624
   470
  by (fastsimp elim!: STL4E [temp_use])
wenzelm@21624
   471
wenzelm@21624
   472
(* "persistently implies infinitely often" *)
wenzelm@21624
   473
lemma DBImplBD: "|- <>[]F --> []<>F"
wenzelm@21624
   474
  apply clarsimp
wenzelm@21624
   475
  apply (rule ccontr)
wenzelm@26305
   476
  apply (simp add: more_temp_simps2)
wenzelm@21624
   477
  apply (drule STL6 [temp_use])
wenzelm@21624
   478
   apply assumption
wenzelm@21624
   479
  apply simp
wenzelm@21624
   480
  done
wenzelm@21624
   481
wenzelm@21624
   482
lemma BoxDmdDmdBox: "|- []<>F & <>[]G --> []<>(F & G)"
wenzelm@21624
   483
  apply clarsimp
wenzelm@21624
   484
  apply (rule ccontr)
wenzelm@26305
   485
  apply (unfold more_temp_simps2)
wenzelm@21624
   486
  apply (drule STL6 [temp_use])
wenzelm@21624
   487
   apply assumption
wenzelm@21624
   488
  apply (subgoal_tac "sigma |= <>[]~F")
wenzelm@21624
   489
   apply (force simp: dmd_def)
wenzelm@21624
   490
  apply (fastsimp elim: DmdImplE [temp_use] STL4E [temp_use])
wenzelm@21624
   491
  done
wenzelm@21624
   492
wenzelm@21624
   493
wenzelm@21624
   494
(* ------------------------------------------------------------------------- *)
wenzelm@21624
   495
(***          TLA-specific theorems: primed formulas                       ***)
wenzelm@21624
   496
(* ------------------------------------------------------------------------- *)
wenzelm@21624
   497
section "priming"
wenzelm@21624
   498
wenzelm@21624
   499
(* ------------------------ TLA2 ------------------------------------------- *)
wenzelm@21624
   500
lemma STL2_pr: "|- []P --> Init P & Init P`"
wenzelm@21624
   501
  by (fastsimp intro!: STL2_gen [temp_use] primeI [temp_use])
wenzelm@21624
   502
wenzelm@21624
   503
(* Auxiliary lemma allows priming of boxed actions *)
wenzelm@21624
   504
lemma BoxPrime: "|- []P --> []($P & P$)"
wenzelm@21624
   505
  apply clarsimp
wenzelm@21624
   506
  apply (erule dup_boxE)
wenzelm@21624
   507
  apply (unfold boxInit_act)
wenzelm@21624
   508
  apply (erule STL4E)
wenzelm@21624
   509
  apply (auto simp: Init_simps dest!: STL2_pr [temp_use])
wenzelm@21624
   510
  done
wenzelm@21624
   511
wenzelm@21624
   512
lemma TLA2:
wenzelm@21624
   513
  assumes "|- $P & P$ --> A"
wenzelm@21624
   514
  shows "|- []P --> []A"
wenzelm@21624
   515
  apply clarsimp
wenzelm@21624
   516
  apply (drule BoxPrime [temp_use])
wenzelm@41529
   517
  apply (auto simp: Init_stp_act_rev [try_rewrite] intro!: assms [temp_use]
wenzelm@21624
   518
    elim!: STL4E [temp_use])
wenzelm@21624
   519
  done
wenzelm@21624
   520
wenzelm@21624
   521
lemma TLA2E: "[| sigma |= []P; |- $P & P$ --> A |] ==> sigma |= []A"
wenzelm@21624
   522
  by (erule (1) TLA2 [temp_use])
wenzelm@21624
   523
wenzelm@21624
   524
lemma DmdPrime: "|- (<>P`) --> (<>P)"
wenzelm@21624
   525
  apply (unfold dmd_def)
wenzelm@21624
   526
  apply (fastsimp elim!: TLA2E [temp_use])
wenzelm@21624
   527
  done
wenzelm@21624
   528
wenzelm@21624
   529
lemmas PrimeDmd = InitDmd_gen [temp_use, THEN DmdPrime [temp_use], standard]
wenzelm@21624
   530
wenzelm@21624
   531
(* ------------------------ INV1, stable --------------------------------------- *)
wenzelm@21624
   532
section "stable, invariant"
wenzelm@21624
   533
wenzelm@21624
   534
lemma ind_rule:
wenzelm@21624
   535
   "[| sigma |= []H; sigma |= Init P; |- H --> (Init P & ~[]F --> Init(P`) & F) |]  
wenzelm@21624
   536
    ==> sigma |= []F"
wenzelm@21624
   537
  apply (rule indT [temp_use])
wenzelm@21624
   538
   apply (erule (2) STL4E)
wenzelm@21624
   539
  done
wenzelm@21624
   540
wenzelm@21624
   541
lemma box_stp_act: "|- ([]$P) = ([]P)"
wenzelm@21624
   542
  by (simp add: boxInit_act Init_simps)
wenzelm@21624
   543
wenzelm@21624
   544
lemmas box_stp_actI = box_stp_act [temp_use, THEN iffD2, standard]
wenzelm@21624
   545
lemmas box_stp_actD = box_stp_act [temp_use, THEN iffD1, standard]
wenzelm@21624
   546
wenzelm@26305
   547
lemmas more_temp_simps3 = box_stp_act [temp_rewrite] more_temp_simps2
wenzelm@21624
   548
wenzelm@21624
   549
lemma INV1: 
wenzelm@21624
   550
  "|- (Init P) --> (stable P) --> []P"
wenzelm@21624
   551
  apply (unfold stable_def boxInit_stp boxInit_act)
wenzelm@21624
   552
  apply clarsimp
wenzelm@21624
   553
  apply (erule ind_rule)
wenzelm@21624
   554
   apply (auto simp: Init_simps elim: ind_rule)
wenzelm@21624
   555
  done
wenzelm@21624
   556
wenzelm@21624
   557
lemma StableT: 
wenzelm@21624
   558
    "!!P. |- $P & A --> P` ==> |- []A --> stable P"
wenzelm@21624
   559
  apply (unfold stable_def)
wenzelm@21624
   560
  apply (fastsimp elim!: STL4E [temp_use])
wenzelm@21624
   561
  done
wenzelm@21624
   562
wenzelm@21624
   563
lemma Stable: "[| sigma |= []A; |- $P & A --> P` |] ==> sigma |= stable P"
wenzelm@21624
   564
  by (erule (1) StableT [temp_use])
wenzelm@21624
   565
wenzelm@21624
   566
(* Generalization of INV1 *)
wenzelm@21624
   567
lemma StableBox: "|- (stable P) --> [](Init P --> []P)"
wenzelm@21624
   568
  apply (unfold stable_def)
wenzelm@21624
   569
  apply clarsimp
wenzelm@21624
   570
  apply (erule dup_boxE)
wenzelm@21624
   571
  apply (force simp: stable_def elim: STL4E [temp_use] INV1 [temp_use])
wenzelm@21624
   572
  done
wenzelm@21624
   573
wenzelm@21624
   574
lemma DmdStable: "|- (stable P) & <>P --> <>[]P"
wenzelm@21624
   575
  apply clarsimp
wenzelm@21624
   576
  apply (rule DmdImpl2)
wenzelm@21624
   577
   prefer 2
wenzelm@21624
   578
   apply (erule StableBox [temp_use])
wenzelm@21624
   579
  apply (simp add: dmdInitD)
wenzelm@21624
   580
  done
wenzelm@21624
   581
wenzelm@21624
   582
(* ---------------- (Semi-)automatic invariant tactics ---------------------- *)
wenzelm@21624
   583
wenzelm@21624
   584
ML {*
wenzelm@21624
   585
(* inv_tac reduces goals of the form ... ==> sigma |= []P *)
wenzelm@42793
   586
fun inv_tac ctxt =
wenzelm@42793
   587
  SELECT_GOAL
wenzelm@42793
   588
    (EVERY
wenzelm@42793
   589
     [auto_tac ctxt,
wenzelm@42793
   590
      TRY (merge_box_tac 1),
wenzelm@42793
   591
      rtac (temp_use @{thm INV1}) 1, (* fail if the goal is not a box *)
wenzelm@42793
   592
      TRYALL (etac @{thm Stable})]);
wenzelm@21624
   593
wenzelm@21624
   594
(* auto_inv_tac applies inv_tac and then tries to attack the subgoals
wenzelm@21624
   595
   in simple cases it may be able to handle goals like |- MyProg --> []Inv.
wenzelm@21624
   596
   In these simple cases the simplifier seems to be more useful than the
wenzelm@21624
   597
   auto-tactic, which applies too much propositional logic and simplifies
wenzelm@21624
   598
   too late.
wenzelm@21624
   599
*)
wenzelm@42793
   600
fun auto_inv_tac ss =
wenzelm@42793
   601
  SELECT_GOAL
wenzelm@42795
   602
    (inv_tac (map_simpset (K ss) @{context}) 1 THEN
wenzelm@42793
   603
      (TRYALL (action_simp_tac
wenzelm@42793
   604
        (ss addsimps [@{thm Init_stp}, @{thm Init_act}]) [] [@{thm squareE}])));
wenzelm@21624
   605
*}
wenzelm@21624
   606
wenzelm@42769
   607
method_setup invariant = {*
wenzelm@42793
   608
  Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o inv_tac))
wenzelm@42769
   609
*} ""
wenzelm@42769
   610
wenzelm@42769
   611
method_setup auto_invariant = {*
wenzelm@42769
   612
  Method.sections Clasimp.clasimp_modifiers
wenzelm@42769
   613
    >> (K (SIMPLE_METHOD' o auto_inv_tac o simpset_of))
wenzelm@42769
   614
*} ""
wenzelm@42769
   615
wenzelm@21624
   616
lemma unless: "|- []($P --> P` | Q`) --> (stable P) | <>Q"
wenzelm@21624
   617
  apply (unfold dmd_def)
wenzelm@21624
   618
  apply (clarsimp dest!: BoxPrime [temp_use])
wenzelm@42787
   619
  apply merge_box
wenzelm@21624
   620
  apply (erule contrapos_np)
wenzelm@21624
   621
  apply (fastsimp elim!: Stable [temp_use])
wenzelm@21624
   622
  done
wenzelm@21624
   623
wenzelm@21624
   624
wenzelm@21624
   625
(* --------------------- Recursive expansions --------------------------------------- *)
wenzelm@21624
   626
section "recursive expansions"
wenzelm@21624
   627
wenzelm@21624
   628
(* Recursive expansions of [] and <> for state predicates *)
wenzelm@21624
   629
lemma BoxRec: "|- ([]P) = (Init P & []P`)"
wenzelm@21624
   630
  apply (auto intro!: STL2_gen [temp_use])
wenzelm@21624
   631
   apply (fastsimp elim!: TLA2E [temp_use])
wenzelm@21624
   632
  apply (auto simp: stable_def elim!: INV1 [temp_use] STL4E [temp_use])
wenzelm@21624
   633
  done
wenzelm@21624
   634
wenzelm@21624
   635
lemma DmdRec: "|- (<>P) = (Init P | <>P`)"
wenzelm@21624
   636
  apply (unfold dmd_def BoxRec [temp_rewrite])
wenzelm@21624
   637
  apply (auto simp: Init_simps)
wenzelm@21624
   638
  done
wenzelm@21624
   639
wenzelm@21624
   640
lemma DmdRec2: "!!sigma. [| sigma |= <>P; sigma |= []~P` |] ==> sigma |= Init P"
wenzelm@21624
   641
  apply (force simp: DmdRec [temp_rewrite] dmd_def)
wenzelm@21624
   642
  done
wenzelm@21624
   643
wenzelm@21624
   644
lemma InfinitePrime: "|- ([]<>P) = ([]<>P`)"
wenzelm@21624
   645
  apply auto
wenzelm@21624
   646
   apply (rule classical)
wenzelm@21624
   647
   apply (rule DBImplBD [temp_use])
wenzelm@21624
   648
   apply (subgoal_tac "sigma |= <>[]P")
wenzelm@21624
   649
    apply (fastsimp elim!: DmdImplE [temp_use] TLA2E [temp_use])
wenzelm@21624
   650
   apply (subgoal_tac "sigma |= <>[] (<>P & []~P`)")
wenzelm@21624
   651
    apply (force simp: boxInit_stp [temp_use]
wenzelm@21624
   652
      elim!: DmdImplE [temp_use] STL4E [temp_use] DmdRec2 [temp_use])
wenzelm@26305
   653
   apply (force intro!: STL6 [temp_use] simp: more_temp_simps3)
wenzelm@21624
   654
  apply (fastsimp intro: DmdPrime [temp_use] elim!: STL4E [temp_use])
wenzelm@21624
   655
  done
wenzelm@21624
   656
wenzelm@21624
   657
lemma InfiniteEnsures:
wenzelm@21624
   658
  "[| sigma |= []N; sigma |= []<>A; |- A & N --> P` |] ==> sigma |= []<>P"
wenzelm@21624
   659
  apply (unfold InfinitePrime [temp_rewrite])
wenzelm@21624
   660
  apply (rule InfImpl)
wenzelm@21624
   661
    apply assumption+
wenzelm@21624
   662
  done
wenzelm@21624
   663
wenzelm@21624
   664
(* ------------------------ fairness ------------------------------------------- *)
wenzelm@21624
   665
section "fairness"
wenzelm@21624
   666
wenzelm@21624
   667
(* alternative definitions of fairness *)
wenzelm@21624
   668
lemma WF_alt: "|- WF(A)_v = ([]<>~Enabled(<A>_v) | []<><A>_v)"
wenzelm@21624
   669
  apply (unfold WF_def dmd_def)
wenzelm@21624
   670
  apply fastsimp
wenzelm@21624
   671
  done
wenzelm@21624
   672
wenzelm@21624
   673
lemma SF_alt: "|- SF(A)_v = (<>[]~Enabled(<A>_v) | []<><A>_v)"
wenzelm@21624
   674
  apply (unfold SF_def dmd_def)
wenzelm@21624
   675
  apply fastsimp
wenzelm@21624
   676
  done
wenzelm@21624
   677
wenzelm@21624
   678
(* theorems to "box" fairness conditions *)
wenzelm@21624
   679
lemma BoxWFI: "|- WF(A)_v --> []WF(A)_v"
wenzelm@26305
   680
  by (auto simp: WF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
wenzelm@21624
   681
wenzelm@21624
   682
lemma WF_Box: "|- ([]WF(A)_v) = WF(A)_v"
wenzelm@21624
   683
  by (fastsimp intro!: BoxWFI [temp_use] dest!: STL2 [temp_use])
wenzelm@21624
   684
wenzelm@21624
   685
lemma BoxSFI: "|- SF(A)_v --> []SF(A)_v"
wenzelm@26305
   686
  by (auto simp: SF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
wenzelm@21624
   687
wenzelm@21624
   688
lemma SF_Box: "|- ([]SF(A)_v) = SF(A)_v"
wenzelm@21624
   689
  by (fastsimp intro!: BoxSFI [temp_use] dest!: STL2 [temp_use])
wenzelm@21624
   690
wenzelm@26305
   691
lemmas more_temp_simps = more_temp_simps3 WF_Box [temp_rewrite] SF_Box [temp_rewrite]
wenzelm@21624
   692
wenzelm@21624
   693
lemma SFImplWF: "|- SF(A)_v --> WF(A)_v"
wenzelm@21624
   694
  apply (unfold SF_def WF_def)
wenzelm@21624
   695
  apply (fastsimp dest!: DBImplBD [temp_use])
wenzelm@21624
   696
  done
wenzelm@21624
   697
wenzelm@21624
   698
(* A tactic that "boxes" all fairness conditions. Apply more_temp_simps to "unbox". *)
wenzelm@21624
   699
ML {*
wenzelm@26305
   700
val box_fair_tac = SELECT_GOAL (REPEAT (dresolve_tac [@{thm BoxWFI}, @{thm BoxSFI}] 1))
wenzelm@21624
   701
*}
wenzelm@21624
   702
wenzelm@21624
   703
wenzelm@21624
   704
(* ------------------------------ leads-to ------------------------------ *)
wenzelm@21624
   705
wenzelm@21624
   706
section "~>"
wenzelm@21624
   707
wenzelm@21624
   708
lemma leadsto_init: "|- (Init F) & (F ~> G) --> <>G"
wenzelm@21624
   709
  apply (unfold leadsto_def)
wenzelm@21624
   710
  apply (auto dest!: STL2 [temp_use])
wenzelm@21624
   711
  done
wenzelm@21624
   712
wenzelm@21624
   713
(* |- F & (F ~> G) --> <>G *)
wenzelm@21624
   714
lemmas leadsto_init_temp = leadsto_init [where 'a = behavior, unfolded Init_simps, standard]
wenzelm@21624
   715
wenzelm@21624
   716
lemma streett_leadsto: "|- ([]<>Init F --> []<>G) = (<>(F ~> G))"
wenzelm@21624
   717
  apply (unfold leadsto_def)
wenzelm@21624
   718
  apply auto
wenzelm@21624
   719
    apply (simp add: more_temp_simps)
wenzelm@21624
   720
    apply (fastsimp elim!: DmdImplE [temp_use] STL4E [temp_use])
wenzelm@21624
   721
   apply (fastsimp intro!: InitDmd [temp_use] elim!: STL4E [temp_use])
wenzelm@21624
   722
  apply (subgoal_tac "sigma |= []<><>G")
wenzelm@21624
   723
   apply (simp add: more_temp_simps)
wenzelm@21624
   724
  apply (drule BoxDmdDmdBox [temp_use])
wenzelm@21624
   725
   apply assumption
wenzelm@21624
   726
  apply (fastsimp elim!: DmdImplE [temp_use] STL4E [temp_use])
wenzelm@21624
   727
  done
wenzelm@21624
   728
wenzelm@21624
   729
lemma leadsto_infinite: "|- []<>F & (F ~> G) --> []<>G"
wenzelm@21624
   730
  apply clarsimp
wenzelm@21624
   731
  apply (erule InitDmd [temp_use, THEN streett_leadsto [temp_unlift, THEN iffD2, THEN mp]])
wenzelm@21624
   732
  apply (simp add: dmdInitD)
wenzelm@21624
   733
  done
wenzelm@21624
   734
wenzelm@21624
   735
(* In particular, strong fairness is a Streett condition. The following
wenzelm@21624
   736
   rules are sometimes easier to use than WF2 or SF2 below.
wenzelm@21624
   737
*)
wenzelm@21624
   738
lemma leadsto_SF: "|- (Enabled(<A>_v) ~> <A>_v) --> SF(A)_v"
wenzelm@21624
   739
  apply (unfold SF_def)
wenzelm@21624
   740
  apply (clarsimp elim!: leadsto_infinite [temp_use])
wenzelm@21624
   741
  done
wenzelm@21624
   742
wenzelm@21624
   743
lemma leadsto_WF: "|- (Enabled(<A>_v) ~> <A>_v) --> WF(A)_v"
wenzelm@21624
   744
  by (clarsimp intro!: SFImplWF [temp_use] leadsto_SF [temp_use])
wenzelm@21624
   745
wenzelm@21624
   746
(* introduce an invariant into the proof of a leadsto assertion.
wenzelm@21624
   747
   []I --> ((P ~> Q)  =  (P /\ I ~> Q))
wenzelm@21624
   748
*)
wenzelm@21624
   749
lemma INV_leadsto: "|- []I & (P & I ~> Q) --> (P ~> Q)"
wenzelm@21624
   750
  apply (unfold leadsto_def)
wenzelm@21624
   751
  apply clarsimp
wenzelm@21624
   752
  apply (erule STL4Edup)
wenzelm@21624
   753
   apply assumption
wenzelm@21624
   754
  apply (auto simp: Init_simps dest!: STL2_gen [temp_use])
wenzelm@21624
   755
  done
wenzelm@21624
   756
wenzelm@21624
   757
lemma leadsto_classical: "|- (Init F & []~G ~> G) --> (F ~> G)"
wenzelm@21624
   758
  apply (unfold leadsto_def dmd_def)
wenzelm@21624
   759
  apply (force simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   760
  done
wenzelm@21624
   761
wenzelm@21624
   762
lemma leadsto_false: "|- (F ~> #False) = ([]~F)"
wenzelm@21624
   763
  apply (unfold leadsto_def)
wenzelm@21624
   764
  apply (simp add: boxNotInitD)
wenzelm@21624
   765
  done
wenzelm@21624
   766
wenzelm@21624
   767
lemma leadsto_exists: "|- ((EX x. F x) ~> G) = (ALL x. (F x ~> G))"
wenzelm@21624
   768
  apply (unfold leadsto_def)
wenzelm@21624
   769
  apply (auto simp: allT [try_rewrite] Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   770
  done
wenzelm@21624
   771
wenzelm@21624
   772
(* basic leadsto properties, cf. Unity *)
wenzelm@21624
   773
wenzelm@21624
   774
lemma ImplLeadsto_gen: "|- [](Init F --> Init G) --> (F ~> G)"
wenzelm@21624
   775
  apply (unfold leadsto_def)
wenzelm@21624
   776
  apply (auto intro!: InitDmd_gen [temp_use]
wenzelm@21624
   777
    elim!: STL4E_gen [temp_use] simp: Init_simps)
wenzelm@21624
   778
  done
wenzelm@21624
   779
wenzelm@21624
   780
lemmas ImplLeadsto = ImplLeadsto_gen [where 'a = behavior and 'b = behavior,
wenzelm@21624
   781
  unfolded Init_simps, standard]
wenzelm@21624
   782
wenzelm@21624
   783
lemma ImplLeadsto_simple: "!!F G. |- F --> G ==> |- F ~> G"
wenzelm@21624
   784
  by (auto simp: Init_def intro!: ImplLeadsto_gen [temp_use] necT [temp_use])
wenzelm@21624
   785
wenzelm@21624
   786
lemma EnsuresLeadsto:
wenzelm@21624
   787
  assumes "|- A & $P --> Q`"
wenzelm@21624
   788
  shows "|- []A --> (P ~> Q)"
wenzelm@21624
   789
  apply (unfold leadsto_def)
wenzelm@21624
   790
  apply (clarsimp elim!: INV_leadsto [temp_use])
wenzelm@21624
   791
  apply (erule STL4E_gen)
wenzelm@21624
   792
  apply (auto simp: Init_defs intro!: PrimeDmd [temp_use] assms [temp_use])
wenzelm@21624
   793
  done
wenzelm@21624
   794
wenzelm@21624
   795
lemma EnsuresLeadsto2: "|- []($P --> Q`) --> (P ~> Q)"
wenzelm@21624
   796
  apply (unfold leadsto_def)
wenzelm@21624
   797
  apply clarsimp
wenzelm@21624
   798
  apply (erule STL4E_gen)
wenzelm@21624
   799
  apply (auto simp: Init_simps intro!: PrimeDmd [temp_use])
wenzelm@21624
   800
  done
wenzelm@21624
   801
wenzelm@21624
   802
lemma ensures:
wenzelm@21624
   803
  assumes 1: "|- $P & N --> P` | Q`"
wenzelm@21624
   804
    and 2: "|- ($P & N) & A --> Q`"
wenzelm@21624
   805
  shows "|- []N & []([]P --> <>A) --> (P ~> Q)"
wenzelm@21624
   806
  apply (unfold leadsto_def)
wenzelm@21624
   807
  apply clarsimp
wenzelm@21624
   808
  apply (erule STL4Edup)
wenzelm@21624
   809
   apply assumption
wenzelm@21624
   810
  apply clarsimp
wenzelm@21624
   811
  apply (subgoal_tac "sigmaa |= [] ($P --> P` | Q`) ")
wenzelm@21624
   812
   apply (drule unless [temp_use])
wenzelm@21624
   813
   apply (clarsimp dest!: INV1 [temp_use])
wenzelm@21624
   814
  apply (rule 2 [THEN DmdImpl, temp_use, THEN DmdPrime [temp_use]])
wenzelm@21624
   815
   apply (force intro!: BoxDmd_simple [temp_use]
wenzelm@21624
   816
     simp: split_box_conj [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
   817
  apply (force elim: STL4E [temp_use] dest: 1 [temp_use])
wenzelm@21624
   818
  done
wenzelm@21624
   819
wenzelm@21624
   820
lemma ensures_simple:
wenzelm@21624
   821
  "[| |- $P & N --> P` | Q`;  
wenzelm@21624
   822
      |- ($P & N) & A --> Q`  
wenzelm@21624
   823
   |] ==> |- []N & []<>A --> (P ~> Q)"
wenzelm@21624
   824
  apply clarsimp
wenzelm@21624
   825
  apply (erule (2) ensures [temp_use])
wenzelm@21624
   826
  apply (force elim!: STL4E [temp_use])
wenzelm@21624
   827
  done
wenzelm@21624
   828
wenzelm@21624
   829
lemma EnsuresInfinite:
wenzelm@21624
   830
    "[| sigma |= []<>P; sigma |= []A; |- A & $P --> Q` |] ==> sigma |= []<>Q"
wenzelm@21624
   831
  apply (erule leadsto_infinite [temp_use])
wenzelm@21624
   832
  apply (erule EnsuresLeadsto [temp_use])
wenzelm@21624
   833
  apply assumption
wenzelm@21624
   834
  done
wenzelm@21624
   835
wenzelm@21624
   836
wenzelm@21624
   837
(*** Gronning's lattice rules (taken from TLP) ***)
wenzelm@21624
   838
section "Lattice rules"
wenzelm@21624
   839
wenzelm@21624
   840
lemma LatticeReflexivity: "|- F ~> F"
wenzelm@21624
   841
  apply (unfold leadsto_def)
wenzelm@21624
   842
  apply (rule necT InitDmd_gen)+
wenzelm@21624
   843
  done
wenzelm@21624
   844
wenzelm@21624
   845
lemma LatticeTransitivity: "|- (G ~> H) & (F ~> G) --> (F ~> H)"
wenzelm@21624
   846
  apply (unfold leadsto_def)
wenzelm@21624
   847
  apply clarsimp
wenzelm@21624
   848
  apply (erule dup_boxE) (* [][] (Init G --> H) *)
wenzelm@42787
   849
  apply merge_box
wenzelm@21624
   850
  apply (clarsimp elim!: STL4E [temp_use])
wenzelm@21624
   851
  apply (rule dup_dmdD)
wenzelm@21624
   852
  apply (subgoal_tac "sigmaa |= <>Init G")
wenzelm@21624
   853
   apply (erule DmdImpl2)
wenzelm@21624
   854
   apply assumption
wenzelm@21624
   855
  apply (simp add: dmdInitD)
wenzelm@21624
   856
  done
wenzelm@21624
   857
wenzelm@21624
   858
lemma LatticeDisjunctionElim1: "|- (F | G ~> H) --> (F ~> H)"
wenzelm@21624
   859
  apply (unfold leadsto_def)
wenzelm@21624
   860
  apply (auto simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   861
  done
wenzelm@21624
   862
wenzelm@21624
   863
lemma LatticeDisjunctionElim2: "|- (F | G ~> H) --> (G ~> H)"
wenzelm@21624
   864
  apply (unfold leadsto_def)
wenzelm@21624
   865
  apply (auto simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   866
  done
wenzelm@21624
   867
wenzelm@21624
   868
lemma LatticeDisjunctionIntro: "|- (F ~> H) & (G ~> H) --> (F | G ~> H)"
wenzelm@21624
   869
  apply (unfold leadsto_def)
wenzelm@21624
   870
  apply clarsimp
wenzelm@42787
   871
  apply merge_box
wenzelm@21624
   872
  apply (auto simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   873
  done
wenzelm@21624
   874
wenzelm@21624
   875
lemma LatticeDisjunction: "|- (F | G ~> H) = ((F ~> H) & (G ~> H))"
wenzelm@21624
   876
  by (auto intro: LatticeDisjunctionIntro [temp_use]
wenzelm@21624
   877
    LatticeDisjunctionElim1 [temp_use]
wenzelm@21624
   878
    LatticeDisjunctionElim2 [temp_use])
wenzelm@21624
   879
wenzelm@21624
   880
lemma LatticeDiamond: "|- (A ~> B | C) & (B ~> D) & (C ~> D) --> (A ~> D)"
wenzelm@21624
   881
  apply clarsimp
wenzelm@21624
   882
  apply (subgoal_tac "sigma |= (B | C) ~> D")
wenzelm@21624
   883
  apply (erule_tac G = "LIFT (B | C)" in LatticeTransitivity [temp_use])
wenzelm@21624
   884
   apply (fastsimp intro!: LatticeDisjunctionIntro [temp_use])+
wenzelm@21624
   885
  done
wenzelm@21624
   886
wenzelm@21624
   887
lemma LatticeTriangle: "|- (A ~> D | B) & (B ~> D) --> (A ~> D)"
wenzelm@21624
   888
  apply clarsimp
wenzelm@21624
   889
  apply (subgoal_tac "sigma |= (D | B) ~> D")
wenzelm@21624
   890
   apply (erule_tac G = "LIFT (D | B)" in LatticeTransitivity [temp_use])
wenzelm@21624
   891
  apply assumption
wenzelm@21624
   892
  apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
wenzelm@21624
   893
  done
wenzelm@21624
   894
wenzelm@21624
   895
lemma LatticeTriangle2: "|- (A ~> B | D) & (B ~> D) --> (A ~> D)"
wenzelm@21624
   896
  apply clarsimp
wenzelm@21624
   897
  apply (subgoal_tac "sigma |= B | D ~> D")
wenzelm@21624
   898
   apply (erule_tac G = "LIFT (B | D)" in LatticeTransitivity [temp_use])
wenzelm@21624
   899
   apply assumption
wenzelm@21624
   900
  apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
wenzelm@21624
   901
  done
wenzelm@21624
   902
wenzelm@21624
   903
(*** Lamport's fairness rules ***)
wenzelm@21624
   904
section "Fairness rules"
wenzelm@21624
   905
wenzelm@21624
   906
lemma WF1:
wenzelm@21624
   907
  "[| |- $P & N  --> P` | Q`;    
wenzelm@21624
   908
      |- ($P & N) & <A>_v --> Q`;    
wenzelm@21624
   909
      |- $P & N --> $(Enabled(<A>_v)) |]    
wenzelm@21624
   910
  ==> |- []N & WF(A)_v --> (P ~> Q)"
wenzelm@21624
   911
  apply (clarsimp dest!: BoxWFI [temp_use])
wenzelm@21624
   912
  apply (erule (2) ensures [temp_use])
wenzelm@21624
   913
  apply (erule (1) STL4Edup)
wenzelm@21624
   914
  apply (clarsimp simp: WF_def)
wenzelm@21624
   915
  apply (rule STL2 [temp_use])
wenzelm@21624
   916
  apply (clarsimp elim!: mp intro!: InitDmd [temp_use])
wenzelm@21624
   917
  apply (erule STL4 [temp_use, THEN box_stp_actD [temp_use]])
wenzelm@21624
   918
  apply (simp add: split_box_conj box_stp_actI)
wenzelm@21624
   919
  done
wenzelm@21624
   920
wenzelm@21624
   921
(* Sometimes easier to use; designed for action B rather than state predicate Q *)
wenzelm@21624
   922
lemma WF_leadsto:
wenzelm@21624
   923
  assumes 1: "|- N & $P --> $Enabled (<A>_v)"
wenzelm@21624
   924
    and 2: "|- N & <A>_v --> B"
wenzelm@21624
   925
    and 3: "|- [](N & [~A]_v) --> stable P"
wenzelm@21624
   926
  shows "|- []N & WF(A)_v --> (P ~> B)"
wenzelm@21624
   927
  apply (unfold leadsto_def)
wenzelm@21624
   928
  apply (clarsimp dest!: BoxWFI [temp_use])
wenzelm@21624
   929
  apply (erule (1) STL4Edup)
wenzelm@21624
   930
  apply clarsimp
wenzelm@21624
   931
  apply (rule 2 [THEN DmdImpl, temp_use])
wenzelm@21624
   932
  apply (rule BoxDmd_simple [temp_use])
wenzelm@21624
   933
   apply assumption
wenzelm@21624
   934
  apply (rule classical)
wenzelm@21624
   935
  apply (rule STL2 [temp_use])
wenzelm@21624
   936
  apply (clarsimp simp: WF_def elim!: mp intro!: InitDmd [temp_use])
wenzelm@21624
   937
  apply (rule 1 [THEN STL4, temp_use, THEN box_stp_actD])
wenzelm@21624
   938
  apply (simp (no_asm_simp) add: split_box_conj [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
   939
  apply (erule INV1 [temp_use])
wenzelm@21624
   940
  apply (rule 3 [temp_use])
wenzelm@21624
   941
  apply (simp add: split_box_conj [try_rewrite] NotDmd [temp_use] not_angle [try_rewrite])
wenzelm@21624
   942
  done
wenzelm@21624
   943
wenzelm@21624
   944
lemma SF1:
wenzelm@21624
   945
  "[| |- $P & N  --> P` | Q`;    
wenzelm@21624
   946
      |- ($P & N) & <A>_v --> Q`;    
wenzelm@21624
   947
      |- []P & []N & []F --> <>Enabled(<A>_v) |]    
wenzelm@21624
   948
  ==> |- []N & SF(A)_v & []F --> (P ~> Q)"
wenzelm@21624
   949
  apply (clarsimp dest!: BoxSFI [temp_use])
wenzelm@21624
   950
  apply (erule (2) ensures [temp_use])
wenzelm@21624
   951
  apply (erule_tac F = F in dup_boxE)
wenzelm@42787
   952
  apply merge_temp_box
wenzelm@21624
   953
  apply (erule STL4Edup)
wenzelm@21624
   954
  apply assumption
wenzelm@21624
   955
  apply (clarsimp simp: SF_def)
wenzelm@21624
   956
  apply (rule STL2 [temp_use])
wenzelm@21624
   957
  apply (erule mp)
wenzelm@21624
   958
  apply (erule STL4 [temp_use])
wenzelm@21624
   959
  apply (simp add: split_box_conj [try_rewrite] STL3 [try_rewrite])
wenzelm@21624
   960
  done
wenzelm@21624
   961
wenzelm@21624
   962
lemma WF2:
wenzelm@21624
   963
  assumes 1: "|- N & <B>_f --> <M>_g"
wenzelm@21624
   964
    and 2: "|- $P & P` & <N & A>_f --> B"
wenzelm@21624
   965
    and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"
wenzelm@21624
   966
    and 4: "|- [](N & [~B]_f) & WF(A)_f & []F & <>[]Enabled(<M>_g) --> <>[]P"
wenzelm@21624
   967
  shows "|- []N & WF(A)_f & []F --> WF(M)_g"
wenzelm@21624
   968
  apply (clarsimp dest!: BoxWFI [temp_use] BoxDmdBox [temp_use, THEN iffD2]
wenzelm@21624
   969
    simp: WF_def [where A = M])
wenzelm@21624
   970
  apply (erule_tac F = F in dup_boxE)
wenzelm@42787
   971
  apply merge_temp_box
wenzelm@21624
   972
  apply (erule STL4Edup)
wenzelm@21624
   973
   apply assumption
wenzelm@21624
   974
  apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
wenzelm@21624
   975
  apply (rule classical)
wenzelm@21624
   976
  apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")
wenzelm@21624
   977
   apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])
wenzelm@21624
   978
  apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])
wenzelm@21624
   979
  apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])
wenzelm@42787
   980
  apply merge_act_box
wenzelm@21624
   981
  apply (frule 4 [temp_use])
wenzelm@21624
   982
     apply assumption+
wenzelm@21624
   983
  apply (drule STL6 [temp_use])
wenzelm@21624
   984
   apply assumption
wenzelm@21624
   985
  apply (erule_tac V = "sigmaa |= <>[]P" in thin_rl)
wenzelm@21624
   986
  apply (erule_tac V = "sigmaa |= []F" in thin_rl)
wenzelm@21624
   987
  apply (drule BoxWFI [temp_use])
wenzelm@21624
   988
  apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)
wenzelm@42787
   989
  apply merge_temp_box
wenzelm@21624
   990
  apply (erule DmdImpldup)
wenzelm@21624
   991
   apply assumption
wenzelm@21624
   992
  apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]
wenzelm@21624
   993
    WF_Box [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
   994
   apply (force elim!: TLA2E [where P = P, temp_use])
wenzelm@21624
   995
  apply (rule STL2 [temp_use])
wenzelm@21624
   996
  apply (force simp: WF_def split_box_conj [try_rewrite]
wenzelm@21624
   997
    elim!: mp intro!: InitDmd [temp_use] 3 [THEN STL4, temp_use])
wenzelm@21624
   998
  done
wenzelm@21624
   999
wenzelm@21624
  1000
lemma SF2:
wenzelm@21624
  1001
  assumes 1: "|- N & <B>_f --> <M>_g"
wenzelm@21624
  1002
    and 2: "|- $P & P` & <N & A>_f --> B"
wenzelm@21624
  1003
    and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"
wenzelm@21624
  1004
    and 4: "|- [](N & [~B]_f) & SF(A)_f & []F & []<>Enabled(<M>_g) --> <>[]P"
wenzelm@21624
  1005
  shows "|- []N & SF(A)_f & []F --> SF(M)_g"
wenzelm@21624
  1006
  apply (clarsimp dest!: BoxSFI [temp_use] simp: 2 [try_rewrite] SF_def [where A = M])
wenzelm@21624
  1007
  apply (erule_tac F = F in dup_boxE)
wenzelm@21624
  1008
  apply (erule_tac F = "TEMP <>Enabled (<M>_g) " in dup_boxE)
wenzelm@42787
  1009
  apply merge_temp_box
wenzelm@21624
  1010
  apply (erule STL4Edup)
wenzelm@21624
  1011
   apply assumption
wenzelm@21624
  1012
  apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
wenzelm@21624
  1013
  apply (rule classical)
wenzelm@21624
  1014
  apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")
wenzelm@21624
  1015
   apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])
wenzelm@21624
  1016
  apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])
wenzelm@21624
  1017
  apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])
wenzelm@42787
  1018
  apply merge_act_box
wenzelm@21624
  1019
  apply (frule 4 [temp_use])
wenzelm@21624
  1020
     apply assumption+
wenzelm@21624
  1021
  apply (erule_tac V = "sigmaa |= []F" in thin_rl)
wenzelm@21624
  1022
  apply (drule BoxSFI [temp_use])
wenzelm@21624
  1023
  apply (erule_tac F = "TEMP <>Enabled (<M>_g)" in dup_boxE)
wenzelm@21624
  1024
  apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)
wenzelm@42787
  1025
  apply merge_temp_box
wenzelm@21624
  1026
  apply (erule DmdImpldup)
wenzelm@21624
  1027
   apply assumption
wenzelm@21624
  1028
  apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]
wenzelm@21624
  1029
    SF_Box [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
  1030
   apply (force elim!: TLA2E [where P = P, temp_use])
wenzelm@21624
  1031
  apply (rule STL2 [temp_use])
wenzelm@21624
  1032
  apply (force simp: SF_def split_box_conj [try_rewrite]
wenzelm@21624
  1033
    elim!: mp InfImpl [temp_use] intro!: 3 [temp_use])
wenzelm@21624
  1034
  done
wenzelm@21624
  1035
wenzelm@21624
  1036
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1037
(***           Liveness proofs by well-founded orderings                   ***)
wenzelm@21624
  1038
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1039
section "Well-founded orderings"
wenzelm@21624
  1040
wenzelm@21624
  1041
lemma wf_leadsto:
wenzelm@21624
  1042
  assumes 1: "wf r"
wenzelm@21624
  1043
    and 2: "!!x. sigma |= F x ~> (G | (EX y. #((y,x):r) & F y))    "
wenzelm@21624
  1044
  shows "sigma |= F x ~> G"
wenzelm@21624
  1045
  apply (rule 1 [THEN wf_induct])
wenzelm@21624
  1046
  apply (rule LatticeTriangle [temp_use])
wenzelm@21624
  1047
   apply (rule 2)
wenzelm@21624
  1048
  apply (auto simp: leadsto_exists [try_rewrite])
wenzelm@21624
  1049
  apply (case_tac "(y,x) :r")
wenzelm@21624
  1050
   apply force
wenzelm@21624
  1051
  apply (force simp: leadsto_def Init_simps intro!: necT [temp_use])
wenzelm@21624
  1052
  done
wenzelm@21624
  1053
wenzelm@21624
  1054
(* If r is well-founded, state function v cannot decrease forever *)
wenzelm@21624
  1055
lemma wf_not_box_decrease: "!!r. wf r ==> |- [][ (v`, $v) : #r ]_v --> <>[][#False]_v"
wenzelm@21624
  1056
  apply clarsimp
wenzelm@21624
  1057
  apply (rule ccontr)
wenzelm@21624
  1058
  apply (subgoal_tac "sigma |= (EX x. v=#x) ~> #False")
wenzelm@21624
  1059
   apply (drule leadsto_false [temp_use, THEN iffD1, THEN STL2_gen [temp_use]])
wenzelm@21624
  1060
   apply (force simp: Init_defs)
wenzelm@21624
  1061
  apply (clarsimp simp: leadsto_exists [try_rewrite] not_square [try_rewrite] more_temp_simps)
wenzelm@21624
  1062
  apply (erule wf_leadsto)
wenzelm@21624
  1063
  apply (rule ensures_simple [temp_use])
wenzelm@21624
  1064
   apply (auto simp: square_def angle_def)
wenzelm@21624
  1065
  done
wenzelm@21624
  1066
wenzelm@21624
  1067
(* "wf r  ==>  |- <>[][ (v`, $v) : #r ]_v --> <>[][#False]_v" *)
wenzelm@21624
  1068
lemmas wf_not_dmd_box_decrease =
wenzelm@21624
  1069
  wf_not_box_decrease [THEN DmdImpl, unfolded more_temp_simps, standard]
wenzelm@21624
  1070
wenzelm@21624
  1071
(* If there are infinitely many steps where v decreases, then there
wenzelm@21624
  1072
   have to be infinitely many non-stuttering steps where v doesn't decrease.
wenzelm@21624
  1073
*)
wenzelm@21624
  1074
lemma wf_box_dmd_decrease:
wenzelm@21624
  1075
  assumes 1: "wf r"
wenzelm@21624
  1076
  shows "|- []<>((v`, $v) : #r) --> []<><(v`, $v) ~: #r>_v"
wenzelm@21624
  1077
  apply clarsimp
wenzelm@21624
  1078
  apply (rule ccontr)
wenzelm@21624
  1079
  apply (simp add: not_angle [try_rewrite] more_temp_simps)
wenzelm@21624
  1080
  apply (drule 1 [THEN wf_not_dmd_box_decrease [temp_use]])
wenzelm@21624
  1081
  apply (drule BoxDmdDmdBox [temp_use])
wenzelm@21624
  1082
   apply assumption
wenzelm@21624
  1083
  apply (subgoal_tac "sigma |= []<> ((#False) ::action)")
wenzelm@21624
  1084
   apply force
wenzelm@21624
  1085
  apply (erule STL4E)
wenzelm@21624
  1086
  apply (rule DmdImpl)
wenzelm@21624
  1087
  apply (force intro: 1 [THEN wf_irrefl, temp_use])
wenzelm@21624
  1088
  done
wenzelm@21624
  1089
wenzelm@21624
  1090
(* In particular, for natural numbers, if n decreases infinitely often
wenzelm@21624
  1091
   then it has to increase infinitely often.
wenzelm@21624
  1092
*)
wenzelm@21624
  1093
lemma nat_box_dmd_decrease: "!!n::nat stfun. |- []<>(n` < $n) --> []<>($n < n`)"
wenzelm@21624
  1094
  apply clarsimp
wenzelm@21624
  1095
  apply (subgoal_tac "sigma |= []<><~ ((n`,$n) : #less_than) >_n")
wenzelm@21624
  1096
   apply (erule thin_rl)
wenzelm@21624
  1097
   apply (erule STL4E)
wenzelm@21624
  1098
   apply (rule DmdImpl)
wenzelm@21624
  1099
   apply (clarsimp simp: angle_def [try_rewrite])
wenzelm@21624
  1100
  apply (rule wf_box_dmd_decrease [temp_use])
wenzelm@21624
  1101
   apply (auto elim!: STL4E [temp_use] DmdImplE [temp_use])
wenzelm@21624
  1102
  done
wenzelm@21624
  1103
wenzelm@21624
  1104
wenzelm@21624
  1105
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1106
(***           Flexible quantification over state variables                ***)
wenzelm@21624
  1107
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1108
section "Flexible quantification"
wenzelm@21624
  1109
wenzelm@21624
  1110
lemma aallI:
wenzelm@21624
  1111
  assumes 1: "basevars vs"
wenzelm@21624
  1112
    and 2: "(!!x. basevars (x,vs) ==> sigma |= F x)"
wenzelm@21624
  1113
  shows "sigma |= (AALL x. F x)"
wenzelm@21624
  1114
  by (auto simp: aall_def elim!: eexE [temp_use] intro!: 1 dest!: 2 [temp_use])
wenzelm@21624
  1115
wenzelm@21624
  1116
lemma aallE: "|- (AALL x. F x) --> F x"
wenzelm@21624
  1117
  apply (unfold aall_def)
wenzelm@21624
  1118
  apply clarsimp
wenzelm@21624
  1119
  apply (erule contrapos_np)
wenzelm@21624
  1120
  apply (force intro!: eexI [temp_use])
wenzelm@21624
  1121
  done
wenzelm@21624
  1122
wenzelm@21624
  1123
(* monotonicity of quantification *)
wenzelm@21624
  1124
lemma eex_mono:
wenzelm@21624
  1125
  assumes 1: "sigma |= EEX x. F x"
wenzelm@21624
  1126
    and 2: "!!x. sigma |= F x --> G x"
wenzelm@21624
  1127
  shows "sigma |= EEX x. G x"
wenzelm@21624
  1128
  apply (rule unit_base [THEN 1 [THEN eexE]])
wenzelm@21624
  1129
  apply (rule eexI [temp_use])
wenzelm@21624
  1130
  apply (erule 2 [unfolded intensional_rews, THEN mp])
wenzelm@21624
  1131
  done
wenzelm@21624
  1132
wenzelm@21624
  1133
lemma aall_mono:
wenzelm@21624
  1134
  assumes 1: "sigma |= AALL x. F(x)"
wenzelm@21624
  1135
    and 2: "!!x. sigma |= F(x) --> G(x)"
wenzelm@21624
  1136
  shows "sigma |= AALL x. G(x)"
wenzelm@21624
  1137
  apply (rule unit_base [THEN aallI])
wenzelm@21624
  1138
  apply (rule 2 [unfolded intensional_rews, THEN mp])
wenzelm@21624
  1139
  apply (rule 1 [THEN aallE [temp_use]])
wenzelm@21624
  1140
  done
wenzelm@21624
  1141
wenzelm@21624
  1142
(* Derived history introduction rule *)
wenzelm@21624
  1143
lemma historyI:
wenzelm@21624
  1144
  assumes 1: "sigma |= Init I"
wenzelm@21624
  1145
    and 2: "sigma |= []N"
wenzelm@21624
  1146
    and 3: "basevars vs"
wenzelm@21624
  1147
    and 4: "!!h. basevars(h,vs) ==> |- I & h = ha --> HI h"
wenzelm@21624
  1148
    and 5: "!!h s t. [| basevars(h,vs); N (s,t); h t = hb (h s) (s,t) |] ==> HN h (s,t)"
wenzelm@21624
  1149
  shows "sigma |= EEX h. Init (HI h) & [](HN h)"
wenzelm@21624
  1150
  apply (rule history [temp_use, THEN eexE])
wenzelm@21624
  1151
  apply (rule 3)
wenzelm@21624
  1152
  apply (rule eexI [temp_use])
wenzelm@21624
  1153
  apply clarsimp
wenzelm@21624
  1154
  apply (rule conjI)
wenzelm@21624
  1155
   prefer 2
wenzelm@21624
  1156
   apply (insert 2)
wenzelm@42787
  1157
   apply merge_box
wenzelm@21624
  1158
   apply (force elim!: STL4E [temp_use] 5 [temp_use])
wenzelm@21624
  1159
  apply (insert 1)
wenzelm@21624
  1160
  apply (force simp: Init_defs elim!: 4 [temp_use])
wenzelm@21624
  1161
  done
wenzelm@21624
  1162
wenzelm@21624
  1163
(* ----------------------------------------------------------------------
wenzelm@21624
  1164
   example of a history variable: existence of a clock
wenzelm@21624
  1165
*)
wenzelm@21624
  1166
wenzelm@21624
  1167
lemma "|- EEX h. Init(h = #True) & [](h` = (~$h))"
wenzelm@21624
  1168
  apply (rule tempI)
wenzelm@21624
  1169
  apply (rule historyI)
wenzelm@21624
  1170
  apply (force simp: Init_defs intro!: unit_base [temp_use] necT [temp_use])+
wenzelm@21624
  1171
  done
wenzelm@21624
  1172
wenzelm@21624
  1173
end