src/HOL/TLA/Action.thy
author blanchet
Tue Nov 07 15:16:42 2017 +0100 (20 months ago)
changeset 67022 49309fe530fd
parent 62146 324bc1ffba12
child 69597 ff784d5a5bfb
permissions -rw-r--r--
more robust parsing for THF proofs (esp. polymorphic Leo-III proofs)
     1 (*  Title:      HOL/TLA/Action.thy
     2     Author:     Stephan Merz
     3     Copyright:  1998 University of Munich
     4 *)
     5 
     6 section \<open>The action level of TLA as an Isabelle theory\<close>
     7 
     8 theory Action
     9 imports Stfun
    10 begin
    11 
    12 type_synonym 'a trfun = "state \<times> state \<Rightarrow> 'a"
    13 type_synonym action = "bool trfun"
    14 
    15 instance prod :: (world, world) world ..
    16 
    17 definition enabled :: "action \<Rightarrow> stpred"
    18   where "enabled A s \<equiv> \<exists>u. (s,u) \<Turnstile> A"
    19 
    20 
    21 consts
    22   before :: "'a stfun \<Rightarrow> 'a trfun"
    23   after :: "'a stfun \<Rightarrow> 'a trfun"
    24   unch :: "'a stfun \<Rightarrow> action"
    25 
    26 syntax
    27   (* Syntax for writing action expressions in arbitrary contexts *)
    28   "_ACT"        :: "lift \<Rightarrow> 'a"                      ("(ACT _)")
    29 
    30   "_before"     :: "lift \<Rightarrow> lift"                    ("($_)"  [100] 99)
    31   "_after"      :: "lift \<Rightarrow> lift"                    ("(_$)"  [100] 99)
    32   "_unchanged"  :: "lift \<Rightarrow> lift"                    ("(unchanged _)" [100] 99)
    33 
    34   (*** Priming: same as "after" ***)
    35   "_prime"      :: "lift \<Rightarrow> lift"                    ("(_`)" [100] 99)
    36 
    37   "_Enabled"    :: "lift \<Rightarrow> lift"                    ("(Enabled _)" [100] 100)
    38 
    39 translations
    40   "ACT A"            =>   "(A::state*state \<Rightarrow> _)"
    41   "_before"          ==   "CONST before"
    42   "_after"           ==   "CONST after"
    43   "_prime"           =>   "_after"
    44   "_unchanged"       ==   "CONST unch"
    45   "_Enabled"         ==   "CONST enabled"
    46   "s \<Turnstile> Enabled A"   <=   "_Enabled A s"
    47   "w \<Turnstile> unchanged f" <=   "_unchanged f w"
    48 
    49 axiomatization where
    50   unl_before:    "(ACT $v) (s,t) \<equiv> v s" and
    51   unl_after:     "(ACT v$) (s,t) \<equiv> v t" and
    52   unchanged_def: "(s,t) \<Turnstile> unchanged v \<equiv> (v t = v s)"
    53 
    54 
    55 definition SqAct :: "[action, 'a stfun] \<Rightarrow> action"
    56   where square_def: "SqAct A v \<equiv> ACT (A \<or> unchanged v)"
    57 
    58 definition AnAct :: "[action, 'a stfun] \<Rightarrow> action"
    59   where angle_def: "AnAct A v \<equiv> ACT (A \<and> \<not> unchanged v)"
    60 
    61 syntax
    62   "_SqAct" :: "[lift, lift] \<Rightarrow> lift"  ("([_]'_(_))" [0, 1000] 99)
    63   "_AnAct" :: "[lift, lift] \<Rightarrow> lift"  ("(<_>'_(_))" [0, 1000] 99)
    64 translations
    65   "_SqAct" == "CONST SqAct"
    66   "_AnAct" == "CONST AnAct"
    67   "w \<Turnstile> [A]_v" \<leftharpoondown> "_SqAct A v w"
    68   "w \<Turnstile> <A>_v" \<leftharpoondown> "_AnAct A v w"
    69 
    70 
    71 (* The following assertion specializes "intI" for any world type
    72    which is a pair, not just for "state * state".
    73 *)
    74 
    75 lemma actionI [intro!]:
    76   assumes "\<And>s t. (s,t) \<Turnstile> A"
    77   shows "\<turnstile> A"
    78   apply (rule assms intI prod.induct)+
    79   done
    80 
    81 lemma actionD [dest]: "\<turnstile> A \<Longrightarrow> (s,t) \<Turnstile> A"
    82   apply (erule intD)
    83   done
    84 
    85 lemma pr_rews [int_rewrite]:
    86   "\<turnstile> (#c)` = #c"
    87   "\<And>f. \<turnstile> f<x>` = f<x` >"
    88   "\<And>f. \<turnstile> f<x,y>` = f<x`,y` >"
    89   "\<And>f. \<turnstile> f<x,y,z>` = f<x`,y`,z` >"
    90   "\<turnstile> (\<forall>x. P x)` = (\<forall>x. (P x)`)"
    91   "\<turnstile> (\<exists>x. P x)` = (\<exists>x. (P x)`)"
    92   by (rule actionI, unfold unl_after intensional_rews, rule refl)+
    93 
    94 
    95 lemmas act_rews [simp] = unl_before unl_after unchanged_def pr_rews
    96 
    97 lemmas action_rews = act_rews intensional_rews
    98 
    99 
   100 (* ================ Functions to "unlift" action theorems into HOL rules ================ *)
   101 
   102 ML \<open>
   103 (* The following functions are specialized versions of the corresponding
   104    functions defined in Intensional.ML in that they introduce a
   105    "world" parameter of the form (s,t) and apply additional rewrites.
   106 *)
   107 
   108 fun action_unlift ctxt th =
   109   (rewrite_rule ctxt @{thms action_rews} (th RS @{thm actionD}))
   110     handle THM _ => int_unlift ctxt th;
   111 
   112 (* Turn  \<turnstile> A = B  into meta-level rewrite rule  A == B *)
   113 val action_rewrite = int_rewrite
   114 
   115 fun action_use ctxt th =
   116     case Thm.concl_of th of
   117       Const _ $ (Const (@{const_name Valid}, _) $ _) =>
   118               (flatten (action_unlift ctxt th) handle THM _ => th)
   119     | _ => th;
   120 \<close>
   121 
   122 attribute_setup action_unlift =
   123   \<open>Scan.succeed (Thm.rule_attribute [] (action_unlift o Context.proof_of))\<close>
   124 attribute_setup action_rewrite =
   125   \<open>Scan.succeed (Thm.rule_attribute [] (action_rewrite o Context.proof_of))\<close>
   126 attribute_setup action_use =
   127   \<open>Scan.succeed (Thm.rule_attribute [] (action_use o Context.proof_of))\<close>
   128 
   129 
   130 (* =========================== square / angle brackets =========================== *)
   131 
   132 lemma idle_squareI: "(s,t) \<Turnstile> unchanged v \<Longrightarrow> (s,t) \<Turnstile> [A]_v"
   133   by (simp add: square_def)
   134 
   135 lemma busy_squareI: "(s,t) \<Turnstile> A \<Longrightarrow> (s,t) \<Turnstile> [A]_v"
   136   by (simp add: square_def)
   137 
   138 lemma squareE [elim]:
   139   "\<lbrakk> (s,t) \<Turnstile> [A]_v; A (s,t) \<Longrightarrow> B (s,t); v t = v s \<Longrightarrow> B (s,t) \<rbrakk> \<Longrightarrow> B (s,t)"
   140   apply (unfold square_def action_rews)
   141   apply (erule disjE)
   142   apply simp_all
   143   done
   144 
   145 lemma squareCI [intro]: "\<lbrakk> v t \<noteq> v s \<Longrightarrow> A (s,t) \<rbrakk> \<Longrightarrow> (s,t) \<Turnstile> [A]_v"
   146   apply (unfold square_def action_rews)
   147   apply (rule disjCI)
   148   apply (erule (1) meta_mp)
   149   done
   150 
   151 lemma angleI [intro]: "\<And>s t. \<lbrakk> A (s,t); v t \<noteq> v s \<rbrakk> \<Longrightarrow> (s,t) \<Turnstile> <A>_v"
   152   by (simp add: angle_def)
   153 
   154 lemma angleE [elim]: "\<lbrakk> (s,t) \<Turnstile> <A>_v; \<lbrakk> A (s,t); v t \<noteq> v s \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
   155   apply (unfold angle_def action_rews)
   156   apply (erule conjE)
   157   apply simp
   158   done
   159 
   160 lemma square_simulation:
   161    "\<And>f. \<lbrakk> \<turnstile> unchanged f \<and> \<not>B \<longrightarrow> unchanged g;
   162             \<turnstile> A \<and> \<not>unchanged g \<longrightarrow> B
   163          \<rbrakk> \<Longrightarrow> \<turnstile> [A]_f \<longrightarrow> [B]_g"
   164   apply clarsimp
   165   apply (erule squareE)
   166   apply (auto simp add: square_def)
   167   done
   168 
   169 lemma not_square: "\<turnstile> (\<not> [A]_v) = <\<not>A>_v"
   170   by (auto simp: square_def angle_def)
   171 
   172 lemma not_angle: "\<turnstile> (\<not> <A>_v) = [\<not>A]_v"
   173   by (auto simp: square_def angle_def)
   174 
   175 
   176 (* ============================== Facts about ENABLED ============================== *)
   177 
   178 lemma enabledI: "\<turnstile> A \<longrightarrow> $Enabled A"
   179   by (auto simp add: enabled_def)
   180 
   181 lemma enabledE: "\<lbrakk> s \<Turnstile> Enabled A; \<And>u. A (s,u) \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> Q"
   182   apply (unfold enabled_def)
   183   apply (erule exE)
   184   apply simp
   185   done
   186 
   187 lemma notEnabledD: "\<turnstile> \<not>$Enabled G \<longrightarrow> \<not> G"
   188   by (auto simp add: enabled_def)
   189 
   190 (* Monotonicity *)
   191 lemma enabled_mono:
   192   assumes min: "s \<Turnstile> Enabled F"
   193     and maj: "\<turnstile> F \<longrightarrow> G"
   194   shows "s \<Turnstile> Enabled G"
   195   apply (rule min [THEN enabledE])
   196   apply (rule enabledI [action_use])
   197   apply (erule maj [action_use])
   198   done
   199 
   200 (* stronger variant *)
   201 lemma enabled_mono2:
   202   assumes min: "s \<Turnstile> Enabled F"
   203     and maj: "\<And>t. F (s,t) \<Longrightarrow> G (s,t)"
   204   shows "s \<Turnstile> Enabled G"
   205   apply (rule min [THEN enabledE])
   206   apply (rule enabledI [action_use])
   207   apply (erule maj)
   208   done
   209 
   210 lemma enabled_disj1: "\<turnstile> Enabled F \<longrightarrow> Enabled (F \<or> G)"
   211   by (auto elim!: enabled_mono)
   212 
   213 lemma enabled_disj2: "\<turnstile> Enabled G \<longrightarrow> Enabled (F \<or> G)"
   214   by (auto elim!: enabled_mono)
   215 
   216 lemma enabled_conj1: "\<turnstile> Enabled (F \<and> G) \<longrightarrow> Enabled F"
   217   by (auto elim!: enabled_mono)
   218 
   219 lemma enabled_conj2: "\<turnstile> Enabled (F \<and> G) \<longrightarrow> Enabled G"
   220   by (auto elim!: enabled_mono)
   221 
   222 lemma enabled_conjE:
   223     "\<lbrakk> s \<Turnstile> Enabled (F \<and> G); \<lbrakk> s \<Turnstile> Enabled F; s \<Turnstile> Enabled G \<rbrakk> \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> Q"
   224   apply (frule enabled_conj1 [action_use])
   225   apply (drule enabled_conj2 [action_use])
   226   apply simp
   227   done
   228 
   229 lemma enabled_disjD: "\<turnstile> Enabled (F \<or> G) \<longrightarrow> Enabled F \<or> Enabled G"
   230   by (auto simp add: enabled_def)
   231 
   232 lemma enabled_disj: "\<turnstile> Enabled (F \<or> G) = (Enabled F \<or> Enabled G)"
   233   apply clarsimp
   234   apply (rule iffI)
   235    apply (erule enabled_disjD [action_use])
   236   apply (erule disjE enabled_disj1 [action_use] enabled_disj2 [action_use])+
   237   done
   238 
   239 lemma enabled_ex: "\<turnstile> Enabled (\<exists>x. F x) = (\<exists>x. Enabled (F x))"
   240   by (force simp add: enabled_def)
   241 
   242 
   243 (* A rule that combines enabledI and baseE, but generates fewer instantiations *)
   244 lemma base_enabled:
   245     "\<lbrakk> basevars vs; \<exists>c. \<forall>u. vs u = c \<longrightarrow> A(s,u) \<rbrakk> \<Longrightarrow> s \<Turnstile> Enabled A"
   246   apply (erule exE)
   247   apply (erule baseE)
   248   apply (rule enabledI [action_use])
   249   apply (erule allE)
   250   apply (erule mp)
   251   apply assumption
   252   done
   253 
   254 (* ======================= action_simp_tac ============================== *)
   255 
   256 ML \<open>
   257 (* A dumb simplification-based tactic with just a little first-order logic:
   258    should plug in only "very safe" rules that can be applied blindly.
   259    Note that it applies whatever simplifications are currently active.
   260 *)
   261 fun action_simp_tac ctxt intros elims =
   262     asm_full_simp_tac
   263          (ctxt setloop (fn _ => (resolve_tac ctxt ((map (action_use ctxt) intros)
   264                                     @ [refl,impI,conjI,@{thm actionI},@{thm intI},allI]))
   265                       ORELSE' (eresolve_tac ctxt ((map (action_use ctxt) elims)
   266                                              @ [conjE,disjE,exE]))));
   267 \<close>
   268 
   269 (* ---------------- enabled_tac: tactic to prove (Enabled A) -------------------- *)
   270 
   271 ML \<open>
   272 (* "Enabled A" can be proven as follows:
   273    - Assume that we know which state variables are "base variables"
   274      this should be expressed by a theorem of the form "basevars (x,y,z,...)".
   275    - Resolve this theorem with baseE to introduce a constant for the value of the
   276      variables in the successor state, and resolve the goal with the result.
   277    - Resolve with enabledI and do some rewriting.
   278    - Solve for the unknowns using standard HOL reasoning.
   279    The following tactic combines these steps except the final one.
   280 *)
   281 
   282 fun enabled_tac ctxt base_vars =
   283   clarsimp_tac (ctxt addSIs [base_vars RS @{thm base_enabled}]);
   284 \<close>
   285 
   286 method_setup enabled = \<open>
   287   Attrib.thm >> (fn th => fn ctxt => SIMPLE_METHOD' (enabled_tac ctxt th))
   288 \<close>
   289 
   290 (* Example *)
   291 
   292 lemma
   293   assumes "basevars (x,y,z)"
   294   shows "\<turnstile> x \<longrightarrow> Enabled ($x \<and> (y$ = #False))"
   295   apply (enabled assms)
   296   apply auto
   297   done
   298 
   299 end