(* Title: HOL/TLA/TLA.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
*)
header {* The temporal level of TLA *}
theory TLA
imports Init
begin
consts
(** abstract syntax **)
Box :: "('w::world) form => temporal"
Dmd :: "('w::world) form => temporal"
leadsto :: "['w::world form, 'v::world form] => temporal"
Stable :: "stpred => temporal"
WF :: "[action, 'a stfun] => temporal"
SF :: "[action, 'a stfun] => temporal"
(* Quantification over (flexible) state variables *)
EEx :: "('a stfun => temporal) => temporal" (binder "Eex " 10)
AAll :: "('a stfun => temporal) => temporal" (binder "Aall " 10)
(** concrete syntax **)
syntax
"_Box" :: "lift => lift" ("([]_)" [40] 40)
"_Dmd" :: "lift => lift" ("(<>_)" [40] 40)
"_leadsto" :: "[lift,lift] => lift" ("(_ ~> _)" [23,22] 22)
"_stable" :: "lift => lift" ("(stable/ _)")
"_WF" :: "[lift,lift] => lift" ("(WF'(_')'_(_))" [0,60] 55)
"_SF" :: "[lift,lift] => lift" ("(SF'(_')'_(_))" [0,60] 55)
"_EEx" :: "[idts, lift] => lift" ("(3EEX _./ _)" [0,10] 10)
"_AAll" :: "[idts, lift] => lift" ("(3AALL _./ _)" [0,10] 10)
translations
"_Box" == "CONST Box"
"_Dmd" == "CONST Dmd"
"_leadsto" == "CONST leadsto"
"_stable" == "CONST Stable"
"_WF" == "CONST WF"
"_SF" == "CONST SF"
"_EEx v A" == "Eex v. A"
"_AAll v A" == "Aall v. A"
"sigma |= []F" <= "_Box F sigma"
"sigma |= <>F" <= "_Dmd F sigma"
"sigma |= F ~> G" <= "_leadsto F G sigma"
"sigma |= stable P" <= "_stable P sigma"
"sigma |= WF(A)_v" <= "_WF A v sigma"
"sigma |= SF(A)_v" <= "_SF A v sigma"
"sigma |= EEX x. F" <= "_EEx x F sigma"
"sigma |= AALL x. F" <= "_AAll x F sigma"
syntax (xsymbols)
"_Box" :: "lift => lift" ("(\<box>_)" [40] 40)
"_Dmd" :: "lift => lift" ("(\<diamond>_)" [40] 40)
"_leadsto" :: "[lift,lift] => lift" ("(_ \<leadsto> _)" [23,22] 22)
"_EEx" :: "[idts, lift] => lift" ("(3\<exists>\<exists> _./ _)" [0,10] 10)
"_AAll" :: "[idts, lift] => lift" ("(3\<forall>\<forall> _./ _)" [0,10] 10)
syntax (HTML output)
"_EEx" :: "[idts, lift] => lift" ("(3\<exists>\<exists> _./ _)" [0,10] 10)
"_AAll" :: "[idts, lift] => lift" ("(3\<forall>\<forall> _./ _)" [0,10] 10)
axiomatization where
(* Definitions of derived operators *)
dmd_def: "\<And>F. TEMP <>F == TEMP ~[]~F"
axiomatization where
boxInit: "\<And>F. TEMP []F == TEMP []Init F" and
leadsto_def: "\<And>F G. TEMP F ~> G == TEMP [](Init F --> <>G)" and
stable_def: "\<And>P. TEMP stable P == TEMP []($P --> P$)" and
WF_def: "TEMP WF(A)_v == TEMP <>[] Enabled(<A>_v) --> []<><A>_v" and
SF_def: "TEMP SF(A)_v == TEMP []<> Enabled(<A>_v) --> []<><A>_v" and
aall_def: "TEMP (AALL x. F x) == TEMP ~ (EEX x. ~ F x)"
axiomatization where
(* Base axioms for raw TLA. *)
normalT: "\<And>F G. |- [](F --> G) --> ([]F --> []G)" and (* polymorphic *)
reflT: "\<And>F. |- []F --> F" and (* F::temporal *)
transT: "\<And>F. |- []F --> [][]F" and (* polymorphic *)
linT: "\<And>F G. |- <>F & <>G --> (<>(F & <>G)) | (<>(G & <>F))" and
discT: "\<And>F. |- [](F --> <>(~F & <>F)) --> (F --> []<>F)" and
primeI: "\<And>P. |- []P --> Init P`" and
primeE: "\<And>P F. |- [](Init P --> []F) --> Init P` --> (F --> []F)" and
indT: "\<And>P F. |- [](Init P & ~[]F --> Init P` & F) --> Init P --> []F" and
allT: "\<And>F. |- (ALL x. [](F x)) = ([](ALL x. F x))"
axiomatization where
necT: "\<And>F. |- F ==> |- []F" (* polymorphic *)
axiomatization where
(* Flexible quantification: refinement mappings, history variables *)
eexI: "|- F x --> (EEX x. F x)" and
eexE: "[| sigma |= (EEX x. F x); basevars vs;
(!!x. [| basevars (x, vs); sigma |= F x |] ==> (G sigma)::bool)
|] ==> G sigma" and
history: "|- EEX h. Init(h = ha) & [](!x. $h = #x --> h` = hb x)"
(* Specialize intensional introduction/elimination rules for temporal formulas *)
lemma tempI [intro!]: "(!!sigma. sigma |= (F::temporal)) ==> |- F"
apply (rule intI)
apply (erule meta_spec)
done
lemma tempD [dest]: "|- (F::temporal) ==> sigma |= F"
by (erule intD)
(* ======== Functions to "unlift" temporal theorems ====== *)
ML {*
(* The following functions are specialized versions of the corresponding
functions defined in theory Intensional in that they introduce a
"world" parameter of type "behavior".
*)
fun temp_unlift th =
(rewrite_rule @{thms action_rews} (th RS @{thm tempD})) handle THM _ => action_unlift th;
(* Turn |- F = G into meta-level rewrite rule F == G *)
val temp_rewrite = int_rewrite
fun temp_use th =
case (concl_of th) of
Const _ $ (Const (@{const_name Intensional.Valid}, _) $ _) =>
((flatten (temp_unlift th)) handle THM _ => th)
| _ => th;
fun try_rewrite th = temp_rewrite th handle THM _ => temp_use th;
*}
attribute_setup temp_unlift = {* Scan.succeed (Thm.rule_attribute (K temp_unlift)) *}
attribute_setup temp_rewrite = {* Scan.succeed (Thm.rule_attribute (K temp_rewrite)) *}
attribute_setup temp_use = {* Scan.succeed (Thm.rule_attribute (K temp_use)) *}
attribute_setup try_rewrite = {* Scan.succeed (Thm.rule_attribute (K try_rewrite)) *}
(* ------------------------------------------------------------------------- *)
(*** "Simple temporal logic": only [] and <> ***)
(* ------------------------------------------------------------------------- *)
section "Simple temporal logic"
(* []~F == []~Init F *)
lemmas boxNotInit = boxInit [of "LIFT ~F", unfolded Init_simps] for F
lemma dmdInit: "TEMP <>F == TEMP <> Init F"
apply (unfold dmd_def)
apply (unfold boxInit [of "LIFT ~F"])
apply (simp (no_asm) add: Init_simps)
done
lemmas dmdNotInit = dmdInit [of "LIFT ~F", unfolded Init_simps] for F
(* boxInit and dmdInit cannot be used as rewrites, because they loop.
Non-looping instances for state predicates and actions are occasionally useful.
*)
lemmas boxInit_stp = boxInit [where 'a = state]
lemmas boxInit_act = boxInit [where 'a = "state * state"]
lemmas dmdInit_stp = dmdInit [where 'a = state]
lemmas dmdInit_act = dmdInit [where 'a = "state * state"]
(* The symmetric equations can be used to get rid of Init *)
lemmas boxInitD = boxInit [symmetric]
lemmas dmdInitD = dmdInit [symmetric]
lemmas boxNotInitD = boxNotInit [symmetric]
lemmas dmdNotInitD = dmdNotInit [symmetric]
lemmas Init_simps = Init_simps boxInitD dmdInitD boxNotInitD dmdNotInitD
(* ------------------------ STL2 ------------------------------------------- *)
lemmas STL2 = reflT
(* The "polymorphic" (generic) variant *)
lemma STL2_gen: "|- []F --> Init F"
apply (unfold boxInit [of F])
apply (rule STL2)
done
(* see also STL2_pr below: "|- []P --> Init P & Init (P`)" *)
(* Dual versions for <> *)
lemma InitDmd: "|- F --> <> F"
apply (unfold dmd_def)
apply (auto dest!: STL2 [temp_use])
done
lemma InitDmd_gen: "|- Init F --> <>F"
apply clarsimp
apply (drule InitDmd [temp_use])
apply (simp add: dmdInitD)
done
(* ------------------------ STL3 ------------------------------------------- *)
lemma STL3: "|- ([][]F) = ([]F)"
by (auto elim: transT [temp_use] STL2 [temp_use])
(* corresponding elimination rule introduces double boxes:
[| (sigma |= []F); (sigma |= [][]F) ==> PROP W |] ==> PROP W
*)
lemmas dup_boxE = STL3 [temp_unlift, THEN iffD2, elim_format]
lemmas dup_boxD = STL3 [temp_unlift, THEN iffD1]
(* dual versions for <> *)
lemma DmdDmd: "|- (<><>F) = (<>F)"
by (auto simp add: dmd_def [try_rewrite] STL3 [try_rewrite])
lemmas dup_dmdE = DmdDmd [temp_unlift, THEN iffD2, elim_format]
lemmas dup_dmdD = DmdDmd [temp_unlift, THEN iffD1]
(* ------------------------ STL4 ------------------------------------------- *)
lemma STL4:
assumes "|- F --> G"
shows "|- []F --> []G"
apply clarsimp
apply (rule normalT [temp_use])
apply (rule assms [THEN necT, temp_use])
apply assumption
done
(* Unlifted version as an elimination rule *)
lemma STL4E: "[| sigma |= []F; |- F --> G |] ==> sigma |= []G"
by (erule (1) STL4 [temp_use])
lemma STL4_gen: "|- Init F --> Init G ==> |- []F --> []G"
apply (drule STL4)
apply (simp add: boxInitD)
done
lemma STL4E_gen: "[| sigma |= []F; |- Init F --> Init G |] ==> sigma |= []G"
by (erule (1) STL4_gen [temp_use])
(* see also STL4Edup below, which allows an auxiliary boxed formula:
[]A /\ F => G
-----------------
[]A /\ []F => []G
*)
(* The dual versions for <> *)
lemma DmdImpl:
assumes prem: "|- F --> G"
shows "|- <>F --> <>G"
apply (unfold dmd_def)
apply (fastforce intro!: prem [temp_use] elim!: STL4E [temp_use])
done
lemma DmdImplE: "[| sigma |= <>F; |- F --> G |] ==> sigma |= <>G"
by (erule (1) DmdImpl [temp_use])
(* ------------------------ STL5 ------------------------------------------- *)
lemma STL5: "|- ([]F & []G) = ([](F & G))"
apply auto
apply (subgoal_tac "sigma |= [] (G --> (F & G))")
apply (erule normalT [temp_use])
apply (fastforce elim!: STL4E [temp_use])+
done
(* rewrite rule to split conjunctions under boxes *)
lemmas split_box_conj = STL5 [temp_unlift, symmetric]
(* the corresponding elimination rule allows to combine boxes in the hypotheses
(NB: F and G must have the same type, i.e., both actions or temporals.)
Use "addSE2" etc. if you want to add this to a claset, otherwise it will loop!
*)
lemma box_conjE:
assumes "sigma |= []F"
and "sigma |= []G"
and "sigma |= [](F&G) ==> PROP R"
shows "PROP R"
by (rule assms STL5 [temp_unlift, THEN iffD1] conjI)+
(* Instances of box_conjE for state predicates, actions, and temporals
in case the general rule is "too polymorphic".
*)
lemmas box_conjE_temp = box_conjE [where 'a = behavior]
lemmas box_conjE_stp = box_conjE [where 'a = state]
lemmas box_conjE_act = box_conjE [where 'a = "state * state"]
(* Define a tactic that tries to merge all boxes in an antecedent. The definition is
a bit kludgy in order to simulate "double elim-resolution".
*)
lemma box_thin: "[| sigma |= []F; PROP W |] ==> PROP W" .
ML {*
fun merge_box_tac i =
REPEAT_DETERM (EVERY [etac @{thm box_conjE} i, atac i, etac @{thm box_thin} i])
fun merge_temp_box_tac ctxt i =
REPEAT_DETERM (EVERY [etac @{thm box_conjE_temp} i, atac i,
eres_inst_tac ctxt [(("'a", 0), "behavior")] @{thm box_thin} i])
fun merge_stp_box_tac ctxt i =
REPEAT_DETERM (EVERY [etac @{thm box_conjE_stp} i, atac i,
eres_inst_tac ctxt [(("'a", 0), "state")] @{thm box_thin} i])
fun merge_act_box_tac ctxt i =
REPEAT_DETERM (EVERY [etac @{thm box_conjE_act} i, atac i,
eres_inst_tac ctxt [(("'a", 0), "state * state")] @{thm box_thin} i])
*}
method_setup merge_box = {* Scan.succeed (K (SIMPLE_METHOD' merge_box_tac)) *}
method_setup merge_temp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_temp_box_tac) *}
method_setup merge_stp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_stp_box_tac) *}
method_setup merge_act_box = {* Scan.succeed (SIMPLE_METHOD' o merge_act_box_tac) *}
(* rewrite rule to push universal quantification through box:
(sigma |= [](! x. F x)) = (! x. (sigma |= []F x))
*)
lemmas all_box = allT [temp_unlift, symmetric]
lemma DmdOr: "|- (<>(F | G)) = (<>F | <>G)"
apply (auto simp add: dmd_def split_box_conj [try_rewrite])
apply (erule contrapos_np, merge_box, fastforce elim!: STL4E [temp_use])+
done
lemma exT: "|- (EX x. <>(F x)) = (<>(EX x. F x))"
by (auto simp: dmd_def Not_Rex [try_rewrite] all_box [try_rewrite])
lemmas ex_dmd = exT [temp_unlift, symmetric]
lemma STL4Edup: "!!sigma. [| sigma |= []A; sigma |= []F; |- F & []A --> G |] ==> sigma |= []G"
apply (erule dup_boxE)
apply merge_box
apply (erule STL4E)
apply assumption
done
lemma DmdImpl2:
"!!sigma. [| sigma |= <>F; sigma |= [](F --> G) |] ==> sigma |= <>G"
apply (unfold dmd_def)
apply auto
apply (erule notE)
apply merge_box
apply (fastforce elim!: STL4E [temp_use])
done
lemma InfImpl:
assumes 1: "sigma |= []<>F"
and 2: "sigma |= []G"
and 3: "|- F & G --> H"
shows "sigma |= []<>H"
apply (insert 1 2)
apply (erule_tac F = G in dup_boxE)
apply merge_box
apply (fastforce elim!: STL4E [temp_use] DmdImpl2 [temp_use] intro!: 3 [temp_use])
done
(* ------------------------ STL6 ------------------------------------------- *)
(* Used in the proof of STL6, but useful in itself. *)
lemma BoxDmd: "|- []F & <>G --> <>([]F & G)"
apply (unfold dmd_def)
apply clarsimp
apply (erule dup_boxE)
apply merge_box
apply (erule contrapos_np)
apply (fastforce elim!: STL4E [temp_use])
done
(* weaker than BoxDmd, but more polymorphic (and often just right) *)
lemma BoxDmd_simple: "|- []F & <>G --> <>(F & G)"
apply (unfold dmd_def)
apply clarsimp
apply merge_box
apply (fastforce elim!: notE STL4E [temp_use])
done
lemma BoxDmd2_simple: "|- []F & <>G --> <>(G & F)"
apply (unfold dmd_def)
apply clarsimp
apply merge_box
apply (fastforce elim!: notE STL4E [temp_use])
done
lemma DmdImpldup:
assumes 1: "sigma |= []A"
and 2: "sigma |= <>F"
and 3: "|- []A & F --> G"
shows "sigma |= <>G"
apply (rule 2 [THEN 1 [THEN BoxDmd [temp_use]], THEN DmdImplE])
apply (rule 3)
done
lemma STL6: "|- <>[]F & <>[]G --> <>[](F & G)"
apply (auto simp: STL5 [temp_rewrite, symmetric])
apply (drule linT [temp_use])
apply assumption
apply (erule thin_rl)
apply (rule DmdDmd [temp_unlift, THEN iffD1])
apply (erule disjE)
apply (erule DmdImplE)
apply (rule BoxDmd)
apply (erule DmdImplE)
apply auto
apply (drule BoxDmd [temp_use])
apply assumption
apply (erule thin_rl)
apply (fastforce elim!: DmdImplE [temp_use])
done
(* ------------------------ True / False ----------------------------------------- *)
section "Simplification of constants"
lemma BoxConst: "|- ([]#P) = #P"
apply (rule tempI)
apply (cases P)
apply (auto intro!: necT [temp_use] dest: STL2_gen [temp_use] simp: Init_simps)
done
lemma DmdConst: "|- (<>#P) = #P"
apply (unfold dmd_def)
apply (cases P)
apply (simp_all add: BoxConst [try_rewrite])
done
lemmas temp_simps [temp_rewrite, simp] = BoxConst DmdConst
(* ------------------------ Further rewrites ----------------------------------------- *)
section "Further rewrites"
lemma NotBox: "|- (~[]F) = (<>~F)"
by (simp add: dmd_def)
lemma NotDmd: "|- (~<>F) = ([]~F)"
by (simp add: dmd_def)
(* These are not declared by default, because they could be harmful,
e.g. []F & ~[]F becomes []F & <>~F !! *)
lemmas more_temp_simps1 =
STL3 [temp_rewrite] DmdDmd [temp_rewrite] NotBox [temp_rewrite] NotDmd [temp_rewrite]
NotBox [temp_unlift, THEN eq_reflection]
NotDmd [temp_unlift, THEN eq_reflection]
lemma BoxDmdBox: "|- ([]<>[]F) = (<>[]F)"
apply (auto dest!: STL2 [temp_use])
apply (rule ccontr)
apply (subgoal_tac "sigma |= <>[][]F & <>[]~[]F")
apply (erule thin_rl)
apply auto
apply (drule STL6 [temp_use])
apply assumption
apply simp
apply (simp_all add: more_temp_simps1)
done
lemma DmdBoxDmd: "|- (<>[]<>F) = ([]<>F)"
apply (unfold dmd_def)
apply (auto simp: BoxDmdBox [unfolded dmd_def, try_rewrite])
done
lemmas more_temp_simps2 = more_temp_simps1 BoxDmdBox [temp_rewrite] DmdBoxDmd [temp_rewrite]
(* ------------------------ Miscellaneous ----------------------------------- *)
lemma BoxOr: "!!sigma. [| sigma |= []F | []G |] ==> sigma |= [](F | G)"
by (fastforce elim!: STL4E [temp_use])
(* "persistently implies infinitely often" *)
lemma DBImplBD: "|- <>[]F --> []<>F"
apply clarsimp
apply (rule ccontr)
apply (simp add: more_temp_simps2)
apply (drule STL6 [temp_use])
apply assumption
apply simp
done
lemma BoxDmdDmdBox: "|- []<>F & <>[]G --> []<>(F & G)"
apply clarsimp
apply (rule ccontr)
apply (unfold more_temp_simps2)
apply (drule STL6 [temp_use])
apply assumption
apply (subgoal_tac "sigma |= <>[]~F")
apply (force simp: dmd_def)
apply (fastforce elim: DmdImplE [temp_use] STL4E [temp_use])
done
(* ------------------------------------------------------------------------- *)
(*** TLA-specific theorems: primed formulas ***)
(* ------------------------------------------------------------------------- *)
section "priming"
(* ------------------------ TLA2 ------------------------------------------- *)
lemma STL2_pr: "|- []P --> Init P & Init P`"
by (fastforce intro!: STL2_gen [temp_use] primeI [temp_use])
(* Auxiliary lemma allows priming of boxed actions *)
lemma BoxPrime: "|- []P --> []($P & P$)"
apply clarsimp
apply (erule dup_boxE)
apply (unfold boxInit_act)
apply (erule STL4E)
apply (auto simp: Init_simps dest!: STL2_pr [temp_use])
done
lemma TLA2:
assumes "|- $P & P$ --> A"
shows "|- []P --> []A"
apply clarsimp
apply (drule BoxPrime [temp_use])
apply (auto simp: Init_stp_act_rev [try_rewrite] intro!: assms [temp_use]
elim!: STL4E [temp_use])
done
lemma TLA2E: "[| sigma |= []P; |- $P & P$ --> A |] ==> sigma |= []A"
by (erule (1) TLA2 [temp_use])
lemma DmdPrime: "|- (<>P`) --> (<>P)"
apply (unfold dmd_def)
apply (fastforce elim!: TLA2E [temp_use])
done
lemmas PrimeDmd = InitDmd_gen [temp_use, THEN DmdPrime [temp_use]]
(* ------------------------ INV1, stable --------------------------------------- *)
section "stable, invariant"
lemma ind_rule:
"[| sigma |= []H; sigma |= Init P; |- H --> (Init P & ~[]F --> Init(P`) & F) |]
==> sigma |= []F"
apply (rule indT [temp_use])
apply (erule (2) STL4E)
done
lemma box_stp_act: "|- ([]$P) = ([]P)"
by (simp add: boxInit_act Init_simps)
lemmas box_stp_actI = box_stp_act [temp_use, THEN iffD2]
lemmas box_stp_actD = box_stp_act [temp_use, THEN iffD1]
lemmas more_temp_simps3 = box_stp_act [temp_rewrite] more_temp_simps2
lemma INV1:
"|- (Init P) --> (stable P) --> []P"
apply (unfold stable_def boxInit_stp boxInit_act)
apply clarsimp
apply (erule ind_rule)
apply (auto simp: Init_simps elim: ind_rule)
done
lemma StableT:
"!!P. |- $P & A --> P` ==> |- []A --> stable P"
apply (unfold stable_def)
apply (fastforce elim!: STL4E [temp_use])
done
lemma Stable: "[| sigma |= []A; |- $P & A --> P` |] ==> sigma |= stable P"
by (erule (1) StableT [temp_use])
(* Generalization of INV1 *)
lemma StableBox: "|- (stable P) --> [](Init P --> []P)"
apply (unfold stable_def)
apply clarsimp
apply (erule dup_boxE)
apply (force simp: stable_def elim: STL4E [temp_use] INV1 [temp_use])
done
lemma DmdStable: "|- (stable P) & <>P --> <>[]P"
apply clarsimp
apply (rule DmdImpl2)
prefer 2
apply (erule StableBox [temp_use])
apply (simp add: dmdInitD)
done
(* ---------------- (Semi-)automatic invariant tactics ---------------------- *)
ML {*
(* inv_tac reduces goals of the form ... ==> sigma |= []P *)
fun inv_tac ctxt =
SELECT_GOAL
(EVERY
[auto_tac ctxt,
TRY (merge_box_tac 1),
rtac (temp_use @{thm INV1}) 1, (* fail if the goal is not a box *)
TRYALL (etac @{thm Stable})]);
(* auto_inv_tac applies inv_tac and then tries to attack the subgoals
in simple cases it may be able to handle goals like |- MyProg --> []Inv.
In these simple cases the simplifier seems to be more useful than the
auto-tactic, which applies too much propositional logic and simplifies
too late.
*)
fun auto_inv_tac ctxt =
SELECT_GOAL
(inv_tac ctxt 1 THEN
(TRYALL (action_simp_tac
(ctxt addsimps [@{thm Init_stp}, @{thm Init_act}]) [] [@{thm squareE}])));
*}
method_setup invariant = {*
Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o inv_tac))
*}
method_setup auto_invariant = {*
Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o auto_inv_tac))
*}
lemma unless: "|- []($P --> P` | Q`) --> (stable P) | <>Q"
apply (unfold dmd_def)
apply (clarsimp dest!: BoxPrime [temp_use])
apply merge_box
apply (erule contrapos_np)
apply (fastforce elim!: Stable [temp_use])
done
(* --------------------- Recursive expansions --------------------------------------- *)
section "recursive expansions"
(* Recursive expansions of [] and <> for state predicates *)
lemma BoxRec: "|- ([]P) = (Init P & []P`)"
apply (auto intro!: STL2_gen [temp_use])
apply (fastforce elim!: TLA2E [temp_use])
apply (auto simp: stable_def elim!: INV1 [temp_use] STL4E [temp_use])
done
lemma DmdRec: "|- (<>P) = (Init P | <>P`)"
apply (unfold dmd_def BoxRec [temp_rewrite])
apply (auto simp: Init_simps)
done
lemma DmdRec2: "!!sigma. [| sigma |= <>P; sigma |= []~P` |] ==> sigma |= Init P"
apply (force simp: DmdRec [temp_rewrite] dmd_def)
done
lemma InfinitePrime: "|- ([]<>P) = ([]<>P`)"
apply auto
apply (rule classical)
apply (rule DBImplBD [temp_use])
apply (subgoal_tac "sigma |= <>[]P")
apply (fastforce elim!: DmdImplE [temp_use] TLA2E [temp_use])
apply (subgoal_tac "sigma |= <>[] (<>P & []~P`)")
apply (force simp: boxInit_stp [temp_use]
elim!: DmdImplE [temp_use] STL4E [temp_use] DmdRec2 [temp_use])
apply (force intro!: STL6 [temp_use] simp: more_temp_simps3)
apply (fastforce intro: DmdPrime [temp_use] elim!: STL4E [temp_use])
done
lemma InfiniteEnsures:
"[| sigma |= []N; sigma |= []<>A; |- A & N --> P` |] ==> sigma |= []<>P"
apply (unfold InfinitePrime [temp_rewrite])
apply (rule InfImpl)
apply assumption+
done
(* ------------------------ fairness ------------------------------------------- *)
section "fairness"
(* alternative definitions of fairness *)
lemma WF_alt: "|- WF(A)_v = ([]<>~Enabled(<A>_v) | []<><A>_v)"
apply (unfold WF_def dmd_def)
apply fastforce
done
lemma SF_alt: "|- SF(A)_v = (<>[]~Enabled(<A>_v) | []<><A>_v)"
apply (unfold SF_def dmd_def)
apply fastforce
done
(* theorems to "box" fairness conditions *)
lemma BoxWFI: "|- WF(A)_v --> []WF(A)_v"
by (auto simp: WF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
lemma WF_Box: "|- ([]WF(A)_v) = WF(A)_v"
by (fastforce intro!: BoxWFI [temp_use] dest!: STL2 [temp_use])
lemma BoxSFI: "|- SF(A)_v --> []SF(A)_v"
by (auto simp: SF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
lemma SF_Box: "|- ([]SF(A)_v) = SF(A)_v"
by (fastforce intro!: BoxSFI [temp_use] dest!: STL2 [temp_use])
lemmas more_temp_simps = more_temp_simps3 WF_Box [temp_rewrite] SF_Box [temp_rewrite]
lemma SFImplWF: "|- SF(A)_v --> WF(A)_v"
apply (unfold SF_def WF_def)
apply (fastforce dest!: DBImplBD [temp_use])
done
(* A tactic that "boxes" all fairness conditions. Apply more_temp_simps to "unbox". *)
ML {*
val box_fair_tac = SELECT_GOAL (REPEAT (dresolve_tac [@{thm BoxWFI}, @{thm BoxSFI}] 1))
*}
(* ------------------------------ leads-to ------------------------------ *)
section "~>"
lemma leadsto_init: "|- (Init F) & (F ~> G) --> <>G"
apply (unfold leadsto_def)
apply (auto dest!: STL2 [temp_use])
done
(* |- F & (F ~> G) --> <>G *)
lemmas leadsto_init_temp = leadsto_init [where 'a = behavior, unfolded Init_simps]
lemma streett_leadsto: "|- ([]<>Init F --> []<>G) = (<>(F ~> G))"
apply (unfold leadsto_def)
apply auto
apply (simp add: more_temp_simps)
apply (fastforce elim!: DmdImplE [temp_use] STL4E [temp_use])
apply (fastforce intro!: InitDmd [temp_use] elim!: STL4E [temp_use])
apply (subgoal_tac "sigma |= []<><>G")
apply (simp add: more_temp_simps)
apply (drule BoxDmdDmdBox [temp_use])
apply assumption
apply (fastforce elim!: DmdImplE [temp_use] STL4E [temp_use])
done
lemma leadsto_infinite: "|- []<>F & (F ~> G) --> []<>G"
apply clarsimp
apply (erule InitDmd [temp_use, THEN streett_leadsto [temp_unlift, THEN iffD2, THEN mp]])
apply (simp add: dmdInitD)
done
(* In particular, strong fairness is a Streett condition. The following
rules are sometimes easier to use than WF2 or SF2 below.
*)
lemma leadsto_SF: "|- (Enabled(<A>_v) ~> <A>_v) --> SF(A)_v"
apply (unfold SF_def)
apply (clarsimp elim!: leadsto_infinite [temp_use])
done
lemma leadsto_WF: "|- (Enabled(<A>_v) ~> <A>_v) --> WF(A)_v"
by (clarsimp intro!: SFImplWF [temp_use] leadsto_SF [temp_use])
(* introduce an invariant into the proof of a leadsto assertion.
[]I --> ((P ~> Q) = (P /\ I ~> Q))
*)
lemma INV_leadsto: "|- []I & (P & I ~> Q) --> (P ~> Q)"
apply (unfold leadsto_def)
apply clarsimp
apply (erule STL4Edup)
apply assumption
apply (auto simp: Init_simps dest!: STL2_gen [temp_use])
done
lemma leadsto_classical: "|- (Init F & []~G ~> G) --> (F ~> G)"
apply (unfold leadsto_def dmd_def)
apply (force simp: Init_simps elim!: STL4E [temp_use])
done
lemma leadsto_false: "|- (F ~> #False) = ([]~F)"
apply (unfold leadsto_def)
apply (simp add: boxNotInitD)
done
lemma leadsto_exists: "|- ((EX x. F x) ~> G) = (ALL x. (F x ~> G))"
apply (unfold leadsto_def)
apply (auto simp: allT [try_rewrite] Init_simps elim!: STL4E [temp_use])
done
(* basic leadsto properties, cf. Unity *)
lemma ImplLeadsto_gen: "|- [](Init F --> Init G) --> (F ~> G)"
apply (unfold leadsto_def)
apply (auto intro!: InitDmd_gen [temp_use]
elim!: STL4E_gen [temp_use] simp: Init_simps)
done
lemmas ImplLeadsto =
ImplLeadsto_gen [where 'a = behavior and 'b = behavior, unfolded Init_simps]
lemma ImplLeadsto_simple: "!!F G. |- F --> G ==> |- F ~> G"
by (auto simp: Init_def intro!: ImplLeadsto_gen [temp_use] necT [temp_use])
lemma EnsuresLeadsto:
assumes "|- A & $P --> Q`"
shows "|- []A --> (P ~> Q)"
apply (unfold leadsto_def)
apply (clarsimp elim!: INV_leadsto [temp_use])
apply (erule STL4E_gen)
apply (auto simp: Init_defs intro!: PrimeDmd [temp_use] assms [temp_use])
done
lemma EnsuresLeadsto2: "|- []($P --> Q`) --> (P ~> Q)"
apply (unfold leadsto_def)
apply clarsimp
apply (erule STL4E_gen)
apply (auto simp: Init_simps intro!: PrimeDmd [temp_use])
done
lemma ensures:
assumes 1: "|- $P & N --> P` | Q`"
and 2: "|- ($P & N) & A --> Q`"
shows "|- []N & []([]P --> <>A) --> (P ~> Q)"
apply (unfold leadsto_def)
apply clarsimp
apply (erule STL4Edup)
apply assumption
apply clarsimp
apply (subgoal_tac "sigmaa |= [] ($P --> P` | Q`) ")
apply (drule unless [temp_use])
apply (clarsimp dest!: INV1 [temp_use])
apply (rule 2 [THEN DmdImpl, temp_use, THEN DmdPrime [temp_use]])
apply (force intro!: BoxDmd_simple [temp_use]
simp: split_box_conj [try_rewrite] box_stp_act [try_rewrite])
apply (force elim: STL4E [temp_use] dest: 1 [temp_use])
done
lemma ensures_simple:
"[| |- $P & N --> P` | Q`;
|- ($P & N) & A --> Q`
|] ==> |- []N & []<>A --> (P ~> Q)"
apply clarsimp
apply (erule (2) ensures [temp_use])
apply (force elim!: STL4E [temp_use])
done
lemma EnsuresInfinite:
"[| sigma |= []<>P; sigma |= []A; |- A & $P --> Q` |] ==> sigma |= []<>Q"
apply (erule leadsto_infinite [temp_use])
apply (erule EnsuresLeadsto [temp_use])
apply assumption
done
(*** Gronning's lattice rules (taken from TLP) ***)
section "Lattice rules"
lemma LatticeReflexivity: "|- F ~> F"
apply (unfold leadsto_def)
apply (rule necT InitDmd_gen)+
done
lemma LatticeTransitivity: "|- (G ~> H) & (F ~> G) --> (F ~> H)"
apply (unfold leadsto_def)
apply clarsimp
apply (erule dup_boxE) (* [][] (Init G --> H) *)
apply merge_box
apply (clarsimp elim!: STL4E [temp_use])
apply (rule dup_dmdD)
apply (subgoal_tac "sigmaa |= <>Init G")
apply (erule DmdImpl2)
apply assumption
apply (simp add: dmdInitD)
done
lemma LatticeDisjunctionElim1: "|- (F | G ~> H) --> (F ~> H)"
apply (unfold leadsto_def)
apply (auto simp: Init_simps elim!: STL4E [temp_use])
done
lemma LatticeDisjunctionElim2: "|- (F | G ~> H) --> (G ~> H)"
apply (unfold leadsto_def)
apply (auto simp: Init_simps elim!: STL4E [temp_use])
done
lemma LatticeDisjunctionIntro: "|- (F ~> H) & (G ~> H) --> (F | G ~> H)"
apply (unfold leadsto_def)
apply clarsimp
apply merge_box
apply (auto simp: Init_simps elim!: STL4E [temp_use])
done
lemma LatticeDisjunction: "|- (F | G ~> H) = ((F ~> H) & (G ~> H))"
by (auto intro: LatticeDisjunctionIntro [temp_use]
LatticeDisjunctionElim1 [temp_use]
LatticeDisjunctionElim2 [temp_use])
lemma LatticeDiamond: "|- (A ~> B | C) & (B ~> D) & (C ~> D) --> (A ~> D)"
apply clarsimp
apply (subgoal_tac "sigma |= (B | C) ~> D")
apply (erule_tac G = "LIFT (B | C)" in LatticeTransitivity [temp_use])
apply (fastforce intro!: LatticeDisjunctionIntro [temp_use])+
done
lemma LatticeTriangle: "|- (A ~> D | B) & (B ~> D) --> (A ~> D)"
apply clarsimp
apply (subgoal_tac "sigma |= (D | B) ~> D")
apply (erule_tac G = "LIFT (D | B)" in LatticeTransitivity [temp_use])
apply assumption
apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
done
lemma LatticeTriangle2: "|- (A ~> B | D) & (B ~> D) --> (A ~> D)"
apply clarsimp
apply (subgoal_tac "sigma |= B | D ~> D")
apply (erule_tac G = "LIFT (B | D)" in LatticeTransitivity [temp_use])
apply assumption
apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
done
(*** Lamport's fairness rules ***)
section "Fairness rules"
lemma WF1:
"[| |- $P & N --> P` | Q`;
|- ($P & N) & <A>_v --> Q`;
|- $P & N --> $(Enabled(<A>_v)) |]
==> |- []N & WF(A)_v --> (P ~> Q)"
apply (clarsimp dest!: BoxWFI [temp_use])
apply (erule (2) ensures [temp_use])
apply (erule (1) STL4Edup)
apply (clarsimp simp: WF_def)
apply (rule STL2 [temp_use])
apply (clarsimp elim!: mp intro!: InitDmd [temp_use])
apply (erule STL4 [temp_use, THEN box_stp_actD [temp_use]])
apply (simp add: split_box_conj box_stp_actI)
done
(* Sometimes easier to use; designed for action B rather than state predicate Q *)
lemma WF_leadsto:
assumes 1: "|- N & $P --> $Enabled (<A>_v)"
and 2: "|- N & <A>_v --> B"
and 3: "|- [](N & [~A]_v) --> stable P"
shows "|- []N & WF(A)_v --> (P ~> B)"
apply (unfold leadsto_def)
apply (clarsimp dest!: BoxWFI [temp_use])
apply (erule (1) STL4Edup)
apply clarsimp
apply (rule 2 [THEN DmdImpl, temp_use])
apply (rule BoxDmd_simple [temp_use])
apply assumption
apply (rule classical)
apply (rule STL2 [temp_use])
apply (clarsimp simp: WF_def elim!: mp intro!: InitDmd [temp_use])
apply (rule 1 [THEN STL4, temp_use, THEN box_stp_actD])
apply (simp (no_asm_simp) add: split_box_conj [try_rewrite] box_stp_act [try_rewrite])
apply (erule INV1 [temp_use])
apply (rule 3 [temp_use])
apply (simp add: split_box_conj [try_rewrite] NotDmd [temp_use] not_angle [try_rewrite])
done
lemma SF1:
"[| |- $P & N --> P` | Q`;
|- ($P & N) & <A>_v --> Q`;
|- []P & []N & []F --> <>Enabled(<A>_v) |]
==> |- []N & SF(A)_v & []F --> (P ~> Q)"
apply (clarsimp dest!: BoxSFI [temp_use])
apply (erule (2) ensures [temp_use])
apply (erule_tac F = F in dup_boxE)
apply merge_temp_box
apply (erule STL4Edup)
apply assumption
apply (clarsimp simp: SF_def)
apply (rule STL2 [temp_use])
apply (erule mp)
apply (erule STL4 [temp_use])
apply (simp add: split_box_conj [try_rewrite] STL3 [try_rewrite])
done
lemma WF2:
assumes 1: "|- N & <B>_f --> <M>_g"
and 2: "|- $P & P` & <N & A>_f --> B"
and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"
and 4: "|- [](N & [~B]_f) & WF(A)_f & []F & <>[]Enabled(<M>_g) --> <>[]P"
shows "|- []N & WF(A)_f & []F --> WF(M)_g"
apply (clarsimp dest!: BoxWFI [temp_use] BoxDmdBox [temp_use, THEN iffD2]
simp: WF_def [where A = M])
apply (erule_tac F = F in dup_boxE)
apply merge_temp_box
apply (erule STL4Edup)
apply assumption
apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
apply (rule classical)
apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")
apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])
apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])
apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])
apply merge_act_box
apply (frule 4 [temp_use])
apply assumption+
apply (drule STL6 [temp_use])
apply assumption
apply (erule_tac V = "sigmaa |= <>[]P" in thin_rl)
apply (erule_tac V = "sigmaa |= []F" in thin_rl)
apply (drule BoxWFI [temp_use])
apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)
apply merge_temp_box
apply (erule DmdImpldup)
apply assumption
apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]
WF_Box [try_rewrite] box_stp_act [try_rewrite])
apply (force elim!: TLA2E [where P = P, temp_use])
apply (rule STL2 [temp_use])
apply (force simp: WF_def split_box_conj [try_rewrite]
elim!: mp intro!: InitDmd [temp_use] 3 [THEN STL4, temp_use])
done
lemma SF2:
assumes 1: "|- N & <B>_f --> <M>_g"
and 2: "|- $P & P` & <N & A>_f --> B"
and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"
and 4: "|- [](N & [~B]_f) & SF(A)_f & []F & []<>Enabled(<M>_g) --> <>[]P"
shows "|- []N & SF(A)_f & []F --> SF(M)_g"
apply (clarsimp dest!: BoxSFI [temp_use] simp: 2 [try_rewrite] SF_def [where A = M])
apply (erule_tac F = F in dup_boxE)
apply (erule_tac F = "TEMP <>Enabled (<M>_g) " in dup_boxE)
apply merge_temp_box
apply (erule STL4Edup)
apply assumption
apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
apply (rule classical)
apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")
apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])
apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])
apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])
apply merge_act_box
apply (frule 4 [temp_use])
apply assumption+
apply (erule_tac V = "sigmaa |= []F" in thin_rl)
apply (drule BoxSFI [temp_use])
apply (erule_tac F = "TEMP <>Enabled (<M>_g)" in dup_boxE)
apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)
apply merge_temp_box
apply (erule DmdImpldup)
apply assumption
apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]
SF_Box [try_rewrite] box_stp_act [try_rewrite])
apply (force elim!: TLA2E [where P = P, temp_use])
apply (rule STL2 [temp_use])
apply (force simp: SF_def split_box_conj [try_rewrite]
elim!: mp InfImpl [temp_use] intro!: 3 [temp_use])
done
(* ------------------------------------------------------------------------- *)
(*** Liveness proofs by well-founded orderings ***)
(* ------------------------------------------------------------------------- *)
section "Well-founded orderings"
lemma wf_leadsto:
assumes 1: "wf r"
and 2: "!!x. sigma |= F x ~> (G | (EX y. #((y,x):r) & F y)) "
shows "sigma |= F x ~> G"
apply (rule 1 [THEN wf_induct])
apply (rule LatticeTriangle [temp_use])
apply (rule 2)
apply (auto simp: leadsto_exists [try_rewrite])
apply (case_tac "(y,x) :r")
apply force
apply (force simp: leadsto_def Init_simps intro!: necT [temp_use])
done
(* If r is well-founded, state function v cannot decrease forever *)
lemma wf_not_box_decrease: "!!r. wf r ==> |- [][ (v`, $v) : #r ]_v --> <>[][#False]_v"
apply clarsimp
apply (rule ccontr)
apply (subgoal_tac "sigma |= (EX x. v=#x) ~> #False")
apply (drule leadsto_false [temp_use, THEN iffD1, THEN STL2_gen [temp_use]])
apply (force simp: Init_defs)
apply (clarsimp simp: leadsto_exists [try_rewrite] not_square [try_rewrite] more_temp_simps)
apply (erule wf_leadsto)
apply (rule ensures_simple [temp_use])
apply (auto simp: square_def angle_def)
done
(* "wf r ==> |- <>[][ (v`, $v) : #r ]_v --> <>[][#False]_v" *)
lemmas wf_not_dmd_box_decrease =
wf_not_box_decrease [THEN DmdImpl, unfolded more_temp_simps]
(* If there are infinitely many steps where v decreases, then there
have to be infinitely many non-stuttering steps where v doesn't decrease.
*)
lemma wf_box_dmd_decrease:
assumes 1: "wf r"
shows "|- []<>((v`, $v) : #r) --> []<><(v`, $v) ~: #r>_v"
apply clarsimp
apply (rule ccontr)
apply (simp add: not_angle [try_rewrite] more_temp_simps)
apply (drule 1 [THEN wf_not_dmd_box_decrease [temp_use]])
apply (drule BoxDmdDmdBox [temp_use])
apply assumption
apply (subgoal_tac "sigma |= []<> ((#False) ::action)")
apply force
apply (erule STL4E)
apply (rule DmdImpl)
apply (force intro: 1 [THEN wf_irrefl, temp_use])
done
(* In particular, for natural numbers, if n decreases infinitely often
then it has to increase infinitely often.
*)
lemma nat_box_dmd_decrease: "!!n::nat stfun. |- []<>(n` < $n) --> []<>($n < n`)"
apply clarsimp
apply (subgoal_tac "sigma |= []<><~ ((n`,$n) : #less_than) >_n")
apply (erule thin_rl)
apply (erule STL4E)
apply (rule DmdImpl)
apply (clarsimp simp: angle_def [try_rewrite])
apply (rule wf_box_dmd_decrease [temp_use])
apply (auto elim!: STL4E [temp_use] DmdImplE [temp_use])
done
(* ------------------------------------------------------------------------- *)
(*** Flexible quantification over state variables ***)
(* ------------------------------------------------------------------------- *)
section "Flexible quantification"
lemma aallI:
assumes 1: "basevars vs"
and 2: "(!!x. basevars (x,vs) ==> sigma |= F x)"
shows "sigma |= (AALL x. F x)"
by (auto simp: aall_def elim!: eexE [temp_use] intro!: 1 dest!: 2 [temp_use])
lemma aallE: "|- (AALL x. F x) --> F x"
apply (unfold aall_def)
apply clarsimp
apply (erule contrapos_np)
apply (force intro!: eexI [temp_use])
done
(* monotonicity of quantification *)
lemma eex_mono:
assumes 1: "sigma |= EEX x. F x"
and 2: "!!x. sigma |= F x --> G x"
shows "sigma |= EEX x. G x"
apply (rule unit_base [THEN 1 [THEN eexE]])
apply (rule eexI [temp_use])
apply (erule 2 [unfolded intensional_rews, THEN mp])
done
lemma aall_mono:
assumes 1: "sigma |= AALL x. F(x)"
and 2: "!!x. sigma |= F(x) --> G(x)"
shows "sigma |= AALL x. G(x)"
apply (rule unit_base [THEN aallI])
apply (rule 2 [unfolded intensional_rews, THEN mp])
apply (rule 1 [THEN aallE [temp_use]])
done
(* Derived history introduction rule *)
lemma historyI:
assumes 1: "sigma |= Init I"
and 2: "sigma |= []N"
and 3: "basevars vs"
and 4: "!!h. basevars(h,vs) ==> |- I & h = ha --> HI h"
and 5: "!!h s t. [| basevars(h,vs); N (s,t); h t = hb (h s) (s,t) |] ==> HN h (s,t)"
shows "sigma |= EEX h. Init (HI h) & [](HN h)"
apply (rule history [temp_use, THEN eexE])
apply (rule 3)
apply (rule eexI [temp_use])
apply clarsimp
apply (rule conjI)
prefer 2
apply (insert 2)
apply merge_box
apply (force elim!: STL4E [temp_use] 5 [temp_use])
apply (insert 1)
apply (force simp: Init_defs elim!: 4 [temp_use])
done
(* ----------------------------------------------------------------------
example of a history variable: existence of a clock
*)
lemma "|- EEX h. Init(h = #True) & [](h` = (~$h))"
apply (rule tempI)
apply (rule historyI)
apply (force simp: Init_defs intro!: unit_base [temp_use] necT [temp_use])+
done
end