src/HOL/TLA/Action.thy
author hoelzl
Tue Nov 05 09:44:57 2013 +0100 (2013-11-05)
changeset 54257 5c7a3b6b05a9
parent 52037 837211662fb8
child 54742 7a86358a3c0b
permissions -rw-r--r--
generalize SUP and INF to the syntactic type classes Sup and Inf
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(*  Title:      HOL/TLA/Action.thy 
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    Author:     Stephan Merz
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    Copyright:  1998 University of Munich
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*)
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header {* The action level of TLA as an Isabelle theory *}
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theory Action
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imports Stfun
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begin
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(** abstract syntax **)
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type_synonym 'a trfun = "(state * state) => 'a"
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type_synonym action   = "bool trfun"
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arities prod :: (world, world) world
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consts
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  (** abstract syntax **)
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  before        :: "'a stfun => 'a trfun"
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  after         :: "'a stfun => 'a trfun"
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  unch          :: "'a stfun => action"
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  SqAct         :: "[action, 'a stfun] => action"
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  AnAct         :: "[action, 'a stfun] => action"
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  enabled       :: "action => stpred"
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(** concrete syntax **)
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syntax
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  (* Syntax for writing action expressions in arbitrary contexts *)
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  "_ACT"        :: "lift => 'a"                      ("(ACT _)")
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  "_before"     :: "lift => lift"                    ("($_)"  [100] 99)
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  "_after"      :: "lift => lift"                    ("(_$)"  [100] 99)
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  "_unchanged"  :: "lift => lift"                    ("(unchanged _)" [100] 99)
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  (*** Priming: same as "after" ***)
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  "_prime"      :: "lift => lift"                    ("(_`)" [100] 99)
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  "_SqAct"      :: "[lift, lift] => lift"            ("([_]'_(_))" [0,1000] 99)
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  "_AnAct"      :: "[lift, lift] => lift"            ("(<_>'_(_))" [0,1000] 99)
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  "_Enabled"    :: "lift => lift"                    ("(Enabled _)" [100] 100)
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translations
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  "ACT A"            =>   "(A::state*state => _)"
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  "_before"          ==   "CONST before"
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  "_after"           ==   "CONST after"
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  "_prime"           =>   "_after"
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  "_unchanged"       ==   "CONST unch"
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  "_SqAct"           ==   "CONST SqAct"
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  "_AnAct"           ==   "CONST AnAct"
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  "_Enabled"         ==   "CONST enabled"
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  "w |= [A]_v"       <=   "_SqAct A v w"
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  "w |= <A>_v"       <=   "_AnAct A v w"
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  "s |= Enabled A"   <=   "_Enabled A s"
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  "w |= unchanged f" <=   "_unchanged f w"
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axiomatization where
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  unl_before:    "(ACT $v) (s,t) == v s" and
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  unl_after:     "(ACT v$) (s,t) == v t" and
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  unchanged_def: "(s,t) |= unchanged v == (v t = v s)"
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defs
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  square_def:    "ACT [A]_v == ACT (A | unchanged v)"
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  angle_def:     "ACT <A>_v == ACT (A & ~ unchanged v)"
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  enabled_def:   "s |= Enabled A  ==  EX u. (s,u) |= A"
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(* The following assertion specializes "intI" for any world type
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   which is a pair, not just for "state * state".
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*)
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lemma actionI [intro!]:
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  assumes "!!s t. (s,t) |= A"
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  shows "|- A"
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  apply (rule assms intI prod.induct)+
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  done
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lemma actionD [dest]: "|- A ==> (s,t) |= A"
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  apply (erule intD)
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  done
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lemma pr_rews [int_rewrite]:
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  "|- (#c)` = #c"
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  "!!f. |- f<x>` = f<x` >"
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  "!!f. |- f<x,y>` = f<x`,y` >"
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  "!!f. |- f<x,y,z>` = f<x`,y`,z` >"
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  "|- (! x. P x)` = (! x. (P x)`)"
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  "|- (? x. P x)` = (? x. (P x)`)"
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  by (rule actionI, unfold unl_after intensional_rews, rule refl)+
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lemmas act_rews [simp] = unl_before unl_after unchanged_def pr_rews
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lemmas action_rews = act_rews intensional_rews
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(* ================ Functions to "unlift" action theorems into HOL rules ================ *)
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ML {*
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(* The following functions are specialized versions of the corresponding
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   functions defined in Intensional.ML in that they introduce a
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   "world" parameter of the form (s,t) and apply additional rewrites.
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*)
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fun action_unlift th =
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  (rewrite_rule @{thms action_rews} (th RS @{thm actionD}))
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    handle THM _ => int_unlift th;
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(* Turn  |- A = B  into meta-level rewrite rule  A == B *)
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val action_rewrite = int_rewrite
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fun action_use th =
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    case (concl_of th) of
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      Const _ $ (Const ("Intensional.Valid", _) $ _) =>
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              (flatten (action_unlift th) handle THM _ => th)
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    | _ => th;
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*}
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attribute_setup action_unlift = {* Scan.succeed (Thm.rule_attribute (K action_unlift)) *}
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attribute_setup action_rewrite = {* Scan.succeed (Thm.rule_attribute (K action_rewrite)) *}
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attribute_setup action_use = {* Scan.succeed (Thm.rule_attribute (K action_use)) *}
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(* =========================== square / angle brackets =========================== *)
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lemma idle_squareI: "(s,t) |= unchanged v ==> (s,t) |= [A]_v"
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  by (simp add: square_def)
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lemma busy_squareI: "(s,t) |= A ==> (s,t) |= [A]_v"
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  by (simp add: square_def)
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lemma squareE [elim]:
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  "[| (s,t) |= [A]_v; A (s,t) ==> B (s,t); v t = v s ==> B (s,t) |] ==> B (s,t)"
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  apply (unfold square_def action_rews)
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  apply (erule disjE)
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  apply simp_all
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  done
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lemma squareCI [intro]: "[| v t ~= v s ==> A (s,t) |] ==> (s,t) |= [A]_v"
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  apply (unfold square_def action_rews)
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  apply (rule disjCI)
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  apply (erule (1) meta_mp)
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  done
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lemma angleI [intro]: "!!s t. [| A (s,t); v t ~= v s |] ==> (s,t) |= <A>_v"
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  by (simp add: angle_def)
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lemma angleE [elim]: "[| (s,t) |= <A>_v; [| A (s,t); v t ~= v s |] ==> R |] ==> R"
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  apply (unfold angle_def action_rews)
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  apply (erule conjE)
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  apply simp
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  done
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lemma square_simulation:
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   "!!f. [| |- unchanged f & ~B --> unchanged g;    
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            |- A & ~unchanged g --> B               
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         |] ==> |- [A]_f --> [B]_g"
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  apply clarsimp
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  apply (erule squareE)
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  apply (auto simp add: square_def)
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  done
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lemma not_square: "|- (~ [A]_v) = <~A>_v"
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  by (auto simp: square_def angle_def)
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lemma not_angle: "|- (~ <A>_v) = [~A]_v"
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  by (auto simp: square_def angle_def)
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(* ============================== Facts about ENABLED ============================== *)
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lemma enabledI: "|- A --> $Enabled A"
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  by (auto simp add: enabled_def)
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lemma enabledE: "[| s |= Enabled A; !!u. A (s,u) ==> Q |] ==> Q"
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  apply (unfold enabled_def)
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  apply (erule exE)
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  apply simp
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  done
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lemma notEnabledD: "|- ~$Enabled G --> ~ G"
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  by (auto simp add: enabled_def)
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(* Monotonicity *)
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lemma enabled_mono:
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  assumes min: "s |= Enabled F"
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    and maj: "|- F --> G"
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  shows "s |= Enabled G"
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  apply (rule min [THEN enabledE])
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  apply (rule enabledI [action_use])
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  apply (erule maj [action_use])
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  done
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(* stronger variant *)
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lemma enabled_mono2:
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  assumes min: "s |= Enabled F"
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    and maj: "!!t. F (s,t) ==> G (s,t)"
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  shows "s |= Enabled G"
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  apply (rule min [THEN enabledE])
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  apply (rule enabledI [action_use])
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  apply (erule maj)
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  done
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lemma enabled_disj1: "|- Enabled F --> Enabled (F | G)"
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  by (auto elim!: enabled_mono)
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lemma enabled_disj2: "|- Enabled G --> Enabled (F | G)"
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  by (auto elim!: enabled_mono)
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lemma enabled_conj1: "|- Enabled (F & G) --> Enabled F"
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  by (auto elim!: enabled_mono)
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lemma enabled_conj2: "|- Enabled (F & G) --> Enabled G"
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  by (auto elim!: enabled_mono)
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lemma enabled_conjE:
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    "[| s |= Enabled (F & G); [| s |= Enabled F; s |= Enabled G |] ==> Q |] ==> Q"
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  apply (frule enabled_conj1 [action_use])
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  apply (drule enabled_conj2 [action_use])
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  apply simp
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  done
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lemma enabled_disjD: "|- Enabled (F | G) --> Enabled F | Enabled G"
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  by (auto simp add: enabled_def)
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lemma enabled_disj: "|- Enabled (F | G) = (Enabled F | Enabled G)"
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  apply clarsimp
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  apply (rule iffI)
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   apply (erule enabled_disjD [action_use])
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  apply (erule disjE enabled_disj1 [action_use] enabled_disj2 [action_use])+
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  done
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lemma enabled_ex: "|- Enabled (EX x. F x) = (EX x. Enabled (F x))"
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  by (force simp add: enabled_def)
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(* A rule that combines enabledI and baseE, but generates fewer instantiations *)
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lemma base_enabled:
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    "[| basevars vs; EX c. ! u. vs u = c --> A(s,u) |] ==> s |= Enabled A"
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  apply (erule exE)
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  apply (erule baseE)
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  apply (rule enabledI [action_use])
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  apply (erule allE)
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  apply (erule mp)
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  apply assumption
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  done
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(* ======================= action_simp_tac ============================== *)
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ML {*
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(* A dumb simplification-based tactic with just a little first-order logic:
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   should plug in only "very safe" rules that can be applied blindly.
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   Note that it applies whatever simplifications are currently active.
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*)
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fun action_simp_tac ss intros elims =
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    asm_full_simp_tac
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         (ss setloop (fn _ => (resolve_tac ((map action_use intros)
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                                    @ [refl,impI,conjI,@{thm actionI},@{thm intI},allI]))
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                      ORELSE' (eresolve_tac ((map action_use elims)
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                                             @ [conjE,disjE,exE]))));
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*}
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(* ---------------- enabled_tac: tactic to prove (Enabled A) -------------------- *)
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ML {*
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(* "Enabled A" can be proven as follows:
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   - Assume that we know which state variables are "base variables"
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     this should be expressed by a theorem of the form "basevars (x,y,z,...)".
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   - Resolve this theorem with baseE to introduce a constant for the value of the
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     variables in the successor state, and resolve the goal with the result.
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   - Resolve with enabledI and do some rewriting.
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   - Solve for the unknowns using standard HOL reasoning.
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   The following tactic combines these steps except the final one.
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*)
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fun enabled_tac ctxt base_vars =
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  clarsimp_tac (ctxt addSIs [base_vars RS @{thm base_enabled}]);
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*}
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method_setup enabled = {*
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  Attrib.thm >> (fn th => fn ctxt => SIMPLE_METHOD' (enabled_tac ctxt th))
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*}
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(* Example *)
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lemma
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  assumes "basevars (x,y,z)"
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  shows "|- x --> Enabled ($x & (y$ = #False))"
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  apply (enabled assms)
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  apply auto
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  done
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end