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