(*
File: TLA/TLA.ML
Author: Stephan Merz
Copyright: 1998 University of Munich
Lemmas and tactics for temporal reasoning.
*)
(* Specialize intensional introduction/elimination rules for temporal formulas *)
qed_goal "tempI" TLA.thy "(!!sigma. sigma |= (F::temporal)) ==> |- F"
(fn [prem] => [ REPEAT (resolve_tac [prem,intI] 1) ]);
qed_goal "tempD" TLA.thy "|- (F::temporal) ==> sigma |= F"
(fn [prem] => [ rtac (prem RS intD) 1 ]);
(* ======== Functions to "unlift" temporal theorems ====== *)
(* The following functions are specialized versions of the corresponding
functions defined in Intensional.ML in that they introduce a
"world" parameter of type "behavior".
*)
fun temp_unlift th =
(rewrite_rule action_rews (th RS tempD))
handle _ => 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 ("Intensional.Valid", _) $ _) =>
((flatten (temp_unlift th)) handle _ => th)
| _ => th;
(* Update classical reasoner---will be updated once more below! *)
AddSIs [tempI];
AddDs [tempD];
val temp_css = (claset(), simpset());
val temp_cs = op addss temp_css;
(* Modify the functions that add rules to simpsets, classical sets,
and clasimpsets in order to accept "lifted" theorems
*)
local
fun try_rewrite th =
(temp_rewrite th) handle _ => temp_use th
in
val op addsimps = fn (ss, ts) => ss addsimps (map try_rewrite ts)
val op addsimps2 = fn (css, ts) => css addsimps2 (map try_rewrite ts)
end;
val op addSIs = fn (cs, ts) => cs addSIs (map temp_use ts);
val op addSEs = fn (cs, ts) => cs addSEs (map temp_use ts);
val op addSDs = fn (cs, ts) => cs addSDs (map temp_use ts);
val op addIs = fn (cs, ts) => cs addIs (map temp_use ts);
val op addEs = fn (cs, ts) => cs addEs (map temp_use ts);
val op addDs = fn (cs, ts) => cs addDs (map temp_use ts);
val op addSIs2 = fn (css, ts) => css addSIs2 (map temp_use ts);
val op addSEs2 = fn (css, ts) => css addSEs2 (map temp_use ts);
val op addSDs2 = fn (css, ts) => css addSDs2 (map temp_use ts);
val op addIs2 = fn (css, ts) => css addIs2 (map temp_use ts);
val op addEs2 = fn (css, ts) => css addEs2 (map temp_use ts);
val op addDs2 = fn (css, ts) => css addDs2 (map temp_use ts);
(* ------------------------------------------------------------------------- *)
(*** "Simple temporal logic": only [] and <> ***)
(* ------------------------------------------------------------------------- *)
section "Simple temporal logic";
(* []~F == []~Init F *)
bind_thm("boxNotInit", rewrite_rule Init_simps (read_instantiate [("F", "LIFT ~F")] boxInit));
qed_goalw "dmdInit" TLA.thy [dmd_def] "TEMP <>F == TEMP <> Init F"
(fn _ => [rewtac (read_instantiate [("F", "LIFT ~F")] boxInit),
simp_tac (simpset() addsimps Init_simps) 1]);
bind_thm("dmdNotInit", rewrite_rule Init_simps (read_instantiate [("F", "LIFT ~F")] dmdInit));
(* boxInit and dmdInit cannot be used as rewrites, because they loop.
Non-looping instances for state predicates and actions are occasionally useful.
*)
bind_thm("boxInit_stp", read_instantiate [("'a","state")] boxInit);
bind_thm("boxInit_act", read_instantiate [("'a","state * state")] boxInit);
bind_thm("dmdInit_stp", read_instantiate [("'a","state")] dmdInit);
bind_thm("dmdInit_act", read_instantiate [("'a","state * state")] dmdInit);
(* The symmetric equations can be used to get rid of Init *)
bind_thm("boxInitD", symmetric boxInit);
bind_thm("dmdInitD", symmetric dmdInit);
bind_thm("boxNotInitD", symmetric boxNotInit);
bind_thm("dmdNotInitD", symmetric dmdNotInit);
val Init_simps = Init_simps @ [boxInitD, dmdInitD, boxNotInitD, dmdNotInitD];
(* ------------------------ STL2 ------------------------------------------- *)
bind_thm("STL2", reflT);
(* The "polymorphic" (generic) variant *)
qed_goal "STL2_gen" TLA.thy "|- []F --> Init F"
(fn _ => [rewtac (read_instantiate [("F", "F")] boxInit),
rtac STL2 1]);
(* see also STL2_pr below: "|- []P --> Init P & Init (P`)" *)
(* Dual versions for <> *)
qed_goalw "InitDmd" TLA.thy [dmd_def] "|- F --> <> F"
(fn _ => [ auto_tac (temp_css addSDs2 [STL2]) ]);
qed_goal "InitDmd_gen" TLA.thy "|- Init F --> <>F"
(fn _ => [Clarsimp_tac 1,
dtac (temp_use InitDmd) 1,
asm_full_simp_tac (simpset() addsimps [dmdInitD]) 1]);
(* ------------------------ STL3 ------------------------------------------- *)
qed_goal "STL3" TLA.thy "|- ([][]F) = ([]F)"
(K [force_tac (temp_css addEs2 [transT,STL2]) 1]);
(* corresponding elimination rule introduces double boxes:
[| (sigma |= []F); (sigma |= [][]F) ==> PROP W |] ==> PROP W
*)
bind_thm("dup_boxE", make_elim((temp_unlift STL3) RS iffD2));
bind_thm("dup_boxD", (temp_unlift STL3) RS iffD1);
(* dual versions for <> *)
qed_goalw "DmdDmd" TLA.thy [dmd_def] "|- (<><>F) = (<>F)"
(fn _ => [ auto_tac (temp_css addsimps2 [STL3]) ]);
bind_thm("dup_dmdE", make_elim((temp_unlift DmdDmd) RS iffD2));
bind_thm("dup_dmdD", (temp_unlift DmdDmd) RS iffD1);
(* ------------------------ STL4 ------------------------------------------- *)
qed_goal "STL4" TLA.thy "|- F --> G ==> |- []F --> []G"
(fn [prem] => [Clarsimp_tac 1,
rtac (temp_use normalT) 1,
rtac (temp_use (prem RS necT)) 1,
atac 1
]);
(* Unlifted version as an elimination rule *)
qed_goal "STL4E" TLA.thy
"[| sigma |= []F; |- F --> G |] ==> sigma |= []G"
(fn prems => [ REPEAT (resolve_tac (prems @ [temp_use STL4]) 1) ]);
qed_goal "STL4_gen" TLA.thy "|- Init F --> Init G ==> |- []F --> []G"
(fn [prem] => [rtac (rewrite_rule [boxInitD] (prem RS STL4)) 1]);
qed_goal "STL4E_gen" TLA.thy
"[| sigma |= []F; |- Init F --> Init G |] ==> sigma |= []G"
(fn prems => [ REPEAT (resolve_tac (prems @ [temp_use STL4_gen]) 1) ]);
(* see also STL4Edup below, which allows an auxiliary boxed formula:
[]A /\ F => G
-----------------
[]A /\ []F => []G
*)
(* The dual versions for <> *)
qed_goalw "DmdImpl" TLA.thy [dmd_def]
"|- F --> G ==> |- <>F --> <>G"
(fn [prem] => [fast_tac (temp_cs addSIs [prem] addSEs [STL4E]) 1]);
qed_goal "DmdImplE" TLA.thy
"[| sigma |= <>F; |- F --> G |] ==> sigma |= <>G"
(fn prems => [ REPEAT (resolve_tac (prems @ [temp_use DmdImpl]) 1) ]);
(* ------------------------ STL5 ------------------------------------------- *)
qed_goal "STL5" TLA.thy "|- ([]F & []G) = ([](F & G))"
(fn _ => [Auto_tac,
subgoal_tac "sigma |= [](G --> (F & G))" 1,
etac (temp_use normalT) 1, atac 1,
ALLGOALS (fast_tac (temp_cs addSEs [STL4E]))
]);
(* rewrite rule to split conjunctions under boxes *)
bind_thm("split_box_conj", (temp_unlift STL5) RS sym);
(* 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!
*)
qed_goal "box_conjE" TLA.thy
"[| sigma |= []F; sigma |= []G; sigma |= [](F&G) ==> PROP R |] ==> PROP R"
(fn prems => [ REPEAT (resolve_tac
(prems @ [(temp_unlift STL5) RS iffD1, conjI]) 1) ]);
(* Instances of box_conjE for state predicates, actions, and temporals
in case the general rule is "too polymorphic".
*)
bind_thm("box_conjE_temp", read_instantiate [("'a","behavior")] box_conjE);
bind_thm("box_conjE_stp", read_instantiate [("'a","state")] box_conjE);
bind_thm("box_conjE_act", read_instantiate [("'a","state * state")] box_conjE);
(* 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".
*)
Goal "[| sigma |= []F; PROP W |] ==> PROP W";
by (atac 1);
val box_thin = result();
fun merge_box_tac i =
REPEAT_DETERM (EVERY [etac box_conjE i, atac i, etac box_thin i]);
fun merge_temp_box_tac i =
REPEAT_DETERM (EVERY [etac box_conjE_temp i, atac i,
eres_inst_tac [("'a","behavior")] box_thin i]);
fun merge_stp_box_tac i =
REPEAT_DETERM (EVERY [etac box_conjE_stp i, atac i,
eres_inst_tac [("'a","state")] box_thin i]);
fun merge_act_box_tac i =
REPEAT_DETERM (EVERY [etac box_conjE_act i, atac i,
eres_inst_tac [("'a","state * state")] box_thin i]);
(* rewrite rule to push universal quantification through box:
(sigma |= [](! x. F x)) = (! x. (sigma |= []F x))
*)
bind_thm("all_box", standard((temp_unlift allT) RS sym));
qed_goal "DmdOr" TLA.thy "|- (<>(F | G)) = (<>F | <>G)"
(fn _ => [auto_tac (temp_css addsimps2 [dmd_def,split_box_conj]),
TRYALL (EVERY' [etac swap,
merge_box_tac,
fast_tac (temp_cs addSEs [STL4E])])
]);
qed_goal "exT" TLA.thy "|- (? x. <>(F x)) = (<>(? x. F x))"
(fn _ => [ auto_tac (temp_css addsimps2 [dmd_def,Not_Rex,all_box]) ]);
bind_thm("ex_dmd", standard((temp_unlift exT) RS sym));
qed_goal "STL4Edup" TLA.thy
"!!sigma. [| sigma |= []A; sigma |= []F; |- F & []A --> G |] ==> sigma |= []G"
(fn _ => [etac dup_boxE 1,
merge_box_tac 1,
etac STL4E 1,
atac 1
]);
qed_goalw "DmdImpl2" TLA.thy [dmd_def]
"!!sigma. [| sigma |= <>F; sigma |= [](F --> G) |] ==> sigma |= <>G"
(fn _ => [Auto_tac,
etac notE 1,
merge_box_tac 1,
fast_tac (temp_cs addSEs [STL4E]) 1
]);
qed_goal "InfImpl" TLA.thy
"[| sigma |= []<>F; sigma |= []G; |- F & G --> H |] ==> sigma |= []<>H"
(fn [prem1,prem2,prem3]
=> [cut_facts_tac [prem1,prem2] 1,
eres_inst_tac [("F","G")] dup_boxE 1,
merge_box_tac 1,
fast_tac (temp_cs addSEs [STL4E,DmdImpl2] addSIs [prem3]) 1
]);
(* ------------------------ STL6 ------------------------------------------- *)
(* Used in the proof of STL6, but useful in itself. *)
qed_goalw "BoxDmd" TLA.thy [dmd_def] "|- []F & <>G --> <>([]F & G)"
(fn _ => [ Clarsimp_tac 1,
etac dup_boxE 1,
merge_box_tac 1,
etac swap 1,
fast_tac (temp_cs addSEs [STL4E]) 1 ]);
(* weaker than BoxDmd, but more polymorphic (and often just right) *)
qed_goalw "BoxDmd_simple" TLA.thy [dmd_def] "|- []F & <>G --> <>(F & G)"
(fn _ => [ Clarsimp_tac 1,
merge_box_tac 1,
fast_tac (temp_cs addSEs [notE,STL4E]) 1
]);
qed_goalw "BoxDmd2_simple" TLA.thy [dmd_def] "|- []F & <>G --> <>(G & F)"
(fn _ => [ Clarsimp_tac 1,
merge_box_tac 1,
fast_tac (temp_cs addSEs [notE,STL4E]) 1
]);
qed_goal "DmdImpldup" TLA.thy
"[| sigma |= []A; sigma |= <>F; |- []A & F --> G |] ==> sigma |= <>G"
(fn [p1,p2,p3] => [rtac ((p2 RS (p1 RS (temp_use BoxDmd))) RS DmdImplE) 1,
rtac p3 1]);
qed_goal "STL6" TLA.thy "|- <>[]F & <>[]G --> <>[](F & G)"
(fn _ => [auto_tac (temp_css addsimps2 [symmetric (temp_rewrite STL5)]),
dtac (temp_use linT) 1, atac 1, etac thin_rl 1,
rtac ((temp_unlift DmdDmd) RS iffD1) 1,
etac disjE 1,
etac DmdImplE 1, rtac BoxDmd 1,
(* the second subgoal needs commutativity of &, which complicates the proof *)
etac DmdImplE 1,
Auto_tac,
dtac (temp_use BoxDmd) 1, atac 1, etac thin_rl 1,
fast_tac (temp_cs addSEs [DmdImplE]) 1
]);
(* ------------------------ True / False ----------------------------------------- *)
section "Simplification of constants";
qed_goal "BoxConst" TLA.thy "|- ([]#P) = #P"
(fn _ => [rtac tempI 1,
case_tac "P" 1,
auto_tac (temp_css addSIs2 [necT] addDs2 [STL2_gen]
addsimps2 Init_simps)
]);
qed_goalw "DmdConst" TLA.thy [dmd_def] "|- (<>#P) = #P"
(fn _ => [case_tac "P" 1,
ALLGOALS (asm_full_simp_tac (simpset() addsimps [BoxConst]))
]);
val temp_simps = map temp_rewrite [BoxConst, DmdConst];
(* Make these rewrites active by default *)
Addsimps temp_simps;
val temp_css = temp_css addsimps2 temp_simps;
val temp_cs = op addss temp_css;
(* ------------------------ Further rewrites ----------------------------------------- *)
section "Further rewrites";
qed_goalw "NotBox" TLA.thy [dmd_def] "|- (~[]F) = (<>~F)"
(fn _ => [ Simp_tac 1 ]);
qed_goalw "NotDmd" TLA.thy [dmd_def] "|- (~<>F) = ([]~F)"
(fn _ => [ Simp_tac 1 ]);
(* These are not by default included in temp_css, because they could be harmful,
e.g. []F & ~[]F becomes []F & <>~F !! *)
val more_temp_simps = (map temp_rewrite [STL3, DmdDmd, NotBox, NotDmd])
@ (map (fn th => (temp_unlift th) RS eq_reflection)
[NotBox, NotDmd]);
qed_goal "BoxDmdBox" TLA.thy "|- ([]<>[]F) = (<>[]F)"
(fn _ => [ auto_tac (temp_css addSDs2 [STL2]),
rtac ccontr 1,
subgoal_tac "sigma |= <>[][]F & <>[]~[]F" 1,
etac thin_rl 1,
Auto_tac,
dtac (temp_use STL6) 1, atac 1,
Asm_full_simp_tac 1,
ALLGOALS (asm_full_simp_tac (simpset() addsimps more_temp_simps))
]);
qed_goalw "DmdBoxDmd" TLA.thy [dmd_def] "|- (<>[]<>F) = ([]<>F)"
(fn _ => [ auto_tac (temp_css addsimps2 [rewrite_rule [dmd_def] BoxDmdBox]) ]);
val more_temp_simps = more_temp_simps @ (map temp_rewrite [BoxDmdBox, DmdBoxDmd]);
(* ------------------------ Miscellaneous ----------------------------------- *)
qed_goal "BoxOr" TLA.thy
"!!sigma. [| sigma |= []F | []G |] ==> sigma |= [](F | G)"
(fn _ => [ fast_tac (temp_cs addSEs [STL4E]) 1 ]);
(* "persistently implies infinitely often" *)
qed_goal "DBImplBD" TLA.thy "|- <>[]F --> []<>F"
(fn _ => [Clarsimp_tac 1,
rtac ccontr 1,
asm_full_simp_tac (simpset() addsimps more_temp_simps) 1,
dtac (temp_use STL6) 1, atac 1,
Asm_full_simp_tac 1
]);
qed_goal "BoxDmdDmdBox" TLA.thy
"|- []<>F & <>[]G --> []<>(F & G)"
(fn _ => [Clarsimp_tac 1,
rtac ccontr 1,
rewrite_goals_tac more_temp_simps,
dtac (temp_use STL6) 1, atac 1,
subgoal_tac "sigma |= <>[]~F" 1,
force_tac (temp_css addsimps2 [dmd_def]) 1,
fast_tac (temp_cs addEs [DmdImplE,STL4E]) 1
]);
(* ------------------------------------------------------------------------- *)
(*** TLA-specific theorems: primed formulas ***)
(* ------------------------------------------------------------------------- *)
section "priming";
(* ------------------------ TLA2 ------------------------------------------- *)
qed_goal "STL2_pr" TLA.thy
"|- []P --> Init P & Init P`"
(fn _ => [fast_tac (temp_cs addSIs [primeI, STL2_gen]) 1]);
(* Auxiliary lemma allows priming of boxed actions *)
qed_goal "BoxPrime" TLA.thy "|- []P --> []($P & P$)"
(fn _ => [Clarsimp_tac 1,
etac dup_boxE 1,
rewtac boxInit_act,
etac STL4E 1,
auto_tac (temp_css addsimps2 Init_simps addSDs2 [STL2_pr])
]);
qed_goal "TLA2" TLA.thy "|- $P & P$ --> A ==> |- []P --> []A"
(fn prems => [Clarsimp_tac 1,
dtac (temp_use BoxPrime) 1,
auto_tac (temp_css addsimps2 [Init_stp_act_rev] addSIs2 prems addSEs2 [STL4E])
]);
qed_goal "TLA2E" TLA.thy
"[| sigma |= []P; |- $P & P$ --> A |] ==> sigma |= []A"
(fn prems => [REPEAT (resolve_tac (prems @ (prems RL [temp_use TLA2])) 1)]);
qed_goalw "DmdPrime" TLA.thy [dmd_def] "|- (<>P`) --> (<>P)"
(fn _ => [ fast_tac (temp_cs addSEs [TLA2E]) 1 ]);
bind_thm("PrimeDmd", (temp_use InitDmd_gen) RS (temp_use DmdPrime));
(* ------------------------ INV1, stable --------------------------------------- *)
section "stable, invariant";
qed_goal "ind_rule" TLA.thy
"[| sigma |= []H; sigma |= Init P; |- H --> (Init P & ~[]F --> Init(P`) & F) |] \
\ ==> sigma |= []F"
(fn prems => [rtac (temp_use indT) 1,
REPEAT (resolve_tac (prems @ (prems RL [STL4E])) 1)]);
qed_goalw "box_stp_act" TLA.thy [boxInit_act] "|- ([]$P) = ([]P)"
(K [simp_tac (simpset() addsimps Init_simps) 1]);
bind_thm("box_stp_actI", zero_var_indexes ((temp_use box_stp_act) RS iffD2));
bind_thm("box_stp_actD", zero_var_indexes ((temp_use box_stp_act) RS iffD1));
val more_temp_simps = (temp_rewrite box_stp_act)::more_temp_simps;
qed_goalw "INV1" TLA.thy [stable_def,boxInit_stp,boxInit_act]
"|- (Init P) --> (stable P) --> []P"
(K [Clarsimp_tac 1,
etac ind_rule 1,
auto_tac (temp_css addsimps2 Init_simps addEs2 [ind_rule])
]);
qed_goalw "StableT" TLA.thy [stable_def]
"|- $P & A --> P` ==> |- []A --> stable P"
(fn [prem] => [fast_tac (temp_cs addSEs [STL4E] addIs [prem]) 1]);
qed_goal "Stable" TLA.thy
"[| sigma |= []A; |- $P & A --> P` |] ==> sigma |= stable P"
(fn prems => [ REPEAT (resolve_tac (prems @ [temp_use StableT]) 1) ]);
(* Generalization of INV1 *)
qed_goalw "StableBox" TLA.thy [stable_def]
"|- (stable P) --> [](Init P --> []P)"
(K [Clarsimp_tac 1,
etac dup_boxE 1,
force_tac (temp_css addsimps2 [stable_def] addEs2 [STL4E, INV1]) 1]);
qed_goal "DmdStable" TLA.thy
"|- (stable P) & <>P --> <>[]P"
(fn _ => [Clarsimp_tac 1,
rtac DmdImpl2 1,
etac (temp_use StableBox) 2,
asm_simp_tac (simpset() addsimps [dmdInitD]) 1
]);
(* ---------------- (Semi-)automatic invariant tactics ---------------------- *)
(* inv_tac reduces goals of the form ... ==> sigma |= []P *)
fun inv_tac css = SELECT_GOAL
(EVERY [auto_tac css,
TRY (merge_box_tac 1),
rtac (temp_use INV1) 1, (* fail if the goal is not a box *)
TRYALL (etac Stable)]);
(* auto_inv_tac applies inv_tac and then tries to attack the subgoals;
in simple cases it may be able to handle goals like |- MyProg --> []Inv.
In these simple cases the simplifier seems to be more useful than the
auto-tactic, which applies too much propositional logic and simplifies
too late.
*)
fun auto_inv_tac ss = SELECT_GOAL
((inv_tac (claset(),ss) 1) THEN
(TRYALL (action_simp_tac (ss addsimps [Init_stp,Init_act]) [] [squareE])));
qed_goalw "unless" TLA.thy [dmd_def]
"|- []($P --> P` | Q`) --> (stable P) | <>Q"
(fn _ => [clarsimp_tac (temp_css addSDs2 [BoxPrime]) 1,
merge_box_tac 1,
etac swap 1,
fast_tac (temp_cs addSEs [Stable]) 1
]);
(* --------------------- Recursive expansions --------------------------------------- *)
section "recursive expansions";
(* Recursive expansions of [] and <> for state predicates *)
qed_goal "BoxRec" TLA.thy "|- ([]P) = (Init P & []P`)"
(fn _ => [auto_tac (temp_css addSIs2 [STL2_gen]),
fast_tac (temp_cs addSEs [TLA2E]) 1,
auto_tac (temp_css addsimps2 [stable_def] addSEs2 [INV1,STL4E])
]);
qed_goalw "DmdRec" TLA.thy [dmd_def, temp_rewrite BoxRec] "|- (<>P) = (Init P | <>P`)"
(K [ auto_tac (temp_css addsimps2 Init_simps) ]);
qed_goal "DmdRec2" TLA.thy
"!!sigma. [| sigma |= <>P; sigma |= []~P` |] ==> sigma |= Init P"
(K [ force_tac (temp_css addsimps2 [DmdRec,dmd_def]) 1]);
(* The "-->" part of the following is a little intricate. *)
qed_goal "InfinitePrime" TLA.thy "|- ([]<>P) = ([]<>P`)"
(fn _ => [Auto_tac,
rtac classical 1,
rtac (temp_use DBImplBD) 1,
subgoal_tac "sigma |= <>[]P" 1,
fast_tac (temp_cs addSEs [DmdImplE,TLA2E]) 1,
subgoal_tac "sigma |= <>[](<>P & []~P`)" 1,
force_tac (temp_css addsimps2 [boxInit_stp]
addSEs2 [DmdImplE,STL4E,DmdRec2]) 1,
force_tac (temp_css addSIs2 [STL6] addsimps2 more_temp_simps) 1,
fast_tac (temp_cs addIs [DmdPrime] addSEs [STL4E]) 1
]);
qed_goal "InfiniteEnsures" TLA.thy
"[| sigma |= []N; sigma |= []<>A; |- A & N --> P` |] ==> sigma |= []<>P"
(fn prems => [rewtac (temp_rewrite InfinitePrime),
rtac InfImpl 1,
REPEAT (resolve_tac prems 1)
]);
(* ------------------------ fairness ------------------------------------------- *)
section "fairness";
(* alternative definitions of fairness *)
qed_goalw "WF_alt" TLA.thy [WF_def,dmd_def]
"|- WF(A)_v = ([]<>~Enabled(<A>_v) | []<><A>_v)"
(fn _ => [ fast_tac temp_cs 1 ]);
qed_goalw "SF_alt" TLA.thy [SF_def,dmd_def]
"|- SF(A)_v = (<>[]~Enabled(<A>_v) | []<><A>_v)"
(fn _ => [ fast_tac temp_cs 1 ]);
(* theorems to "box" fairness conditions *)
qed_goal "BoxWFI" TLA.thy "|- WF(A)_v --> []WF(A)_v"
(fn _ => [ auto_tac (temp_css addsimps2 (WF_alt::more_temp_simps)
addSIs2 [BoxOr]) ]);
qed_goal "WF_Box" TLA.thy "|- ([]WF(A)_v) = WF(A)_v"
(fn prems => [ fast_tac (temp_cs addSIs [BoxWFI] addSDs [STL2]) 1 ]);
qed_goal "BoxSFI" TLA.thy "|- SF(A)_v --> []SF(A)_v"
(fn _ => [ auto_tac (temp_css addsimps2 (SF_alt::more_temp_simps)
addSIs2 [BoxOr]) ]);
qed_goal "SF_Box" TLA.thy "|- ([]SF(A)_v) = SF(A)_v"
(fn prems => [ fast_tac (temp_cs addSIs [BoxSFI] addSDs [STL2]) 1 ]);
val more_temp_simps = more_temp_simps @ (map temp_rewrite [WF_Box, SF_Box]);
qed_goalw "SFImplWF" TLA.thy [SF_def,WF_def] "|- SF(A)_v --> WF(A)_v"
(fn _ => [ fast_tac (temp_cs addSDs [DBImplBD]) 1 ]);
(* A tactic that "boxes" all fairness conditions. Apply more_temp_simps to "unbox". *)
val box_fair_tac = SELECT_GOAL (REPEAT (dresolve_tac [BoxWFI,BoxSFI] 1));
(* ------------------------------ leads-to ------------------------------ *)
section "~>";
qed_goalw "leadsto_init" TLA.thy [leadsto_def]
"|- (Init F) & (F ~> G) --> <>G"
(fn _ => [ auto_tac (temp_css addSDs2 [STL2]) ]);
(* |- F & (F ~> G) --> <>G *)
bind_thm("leadsto_init_temp",
rewrite_rule Init_simps (read_instantiate [("'a","behavior")] leadsto_init));
qed_goalw "streett_leadsto" TLA.thy [leadsto_def]
"|- ([]<>Init F --> []<>G) = (<>(F ~> G))" (K [
Auto_tac,
asm_full_simp_tac (simpset() addsimps more_temp_simps) 1,
fast_tac (temp_cs addSEs [DmdImplE,STL4E]) 1,
fast_tac (temp_cs addSIs [InitDmd] addSEs [STL4E]) 1,
subgoal_tac "sigma |= []<><>G" 1,
asm_full_simp_tac (simpset() addsimps more_temp_simps) 1,
dtac (temp_use BoxDmdDmdBox) 1, atac 1,
fast_tac (temp_cs addSEs [DmdImplE,STL4E]) 1
]);
qed_goal "leadsto_infinite" TLA.thy
"|- []<>F & (F ~> G) --> []<>G"
(fn _ => [Clarsimp_tac 1,
etac ((temp_use InitDmd) RS
((temp_unlift streett_leadsto) RS iffD2 RS mp)) 1,
asm_simp_tac (simpset() addsimps [dmdInitD]) 1
]);
(* In particular, strong fairness is a Streett condition. The following
rules are sometimes easier to use than WF2 or SF2 below.
*)
qed_goalw "leadsto_SF" TLA.thy [SF_def]
"|- (Enabled(<A>_v) ~> <A>_v) --> SF(A)_v"
(K [clarsimp_tac (temp_css addSEs2 [leadsto_infinite]) 1]);
qed_goal "leadsto_WF" TLA.thy
"|- (Enabled(<A>_v) ~> <A>_v) --> WF(A)_v"
(K [ clarsimp_tac (temp_css addSIs2 [SFImplWF, leadsto_SF]) 1 ]);
(* introduce an invariant into the proof of a leadsto assertion.
[]I --> ((P ~> Q) = (P /\ I ~> Q))
*)
qed_goalw "INV_leadsto" TLA.thy [leadsto_def]
"|- []I & (P & I ~> Q) --> (P ~> Q)"
(fn _ => [Clarsimp_tac 1,
etac STL4Edup 1, atac 1,
auto_tac (temp_css addsimps2 Init_simps addSDs2 [STL2_gen])
]);
qed_goalw "leadsto_classical" TLA.thy [leadsto_def,dmd_def]
"|- (Init F & []~G ~> G) --> (F ~> G)"
(fn _ => [force_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E]) 1]);
qed_goalw "leadsto_false" TLA.thy [leadsto_def]
"|- (F ~> #False) = ([]~F)"
(fn _ => [ simp_tac (simpset() addsimps [boxNotInitD]) 1 ]);
qed_goalw "leadsto_exists" TLA.thy [leadsto_def]
"|- ((? x. F x) ~> G) = (!x. (F x ~> G))"
(K [auto_tac (temp_css addsimps2 allT::Init_simps addSEs2 [STL4E])]);
(* basic leadsto properties, cf. Unity *)
qed_goalw "ImplLeadsto_gen" TLA.thy [leadsto_def]
"|- [](Init F --> Init G) --> (F ~> G)"
(fn _ => [auto_tac (temp_css addSIs2 [InitDmd_gen]
addSEs2 [STL4E_gen] addsimps2 Init_simps)
]);
bind_thm("ImplLeadsto",
rewrite_rule Init_simps
(read_instantiate [("'a","behavior"), ("'b","behavior")] ImplLeadsto_gen));
qed_goal "ImplLeadsto_simple" TLA.thy
"|- F --> G ==> |- F ~> G"
(fn [prem] => [auto_tac (temp_css addsimps2 [Init_def]
addSIs2 [ImplLeadsto_gen,necT,prem])]);
qed_goalw "EnsuresLeadsto" TLA.thy [leadsto_def]
"|- A & $P --> Q` ==> |- []A --> (P ~> Q)" (fn [prem] => [
clarsimp_tac (temp_css addSEs2 [INV_leadsto]) 1,
etac STL4E_gen 1,
auto_tac (temp_css addsimps2 Init_defs
addSIs2 [PrimeDmd, prem])
]);
qed_goalw "EnsuresLeadsto2" TLA.thy [leadsto_def]
"|- []($P --> Q`) --> (P ~> Q)"
(fn _ => [Clarsimp_tac 1,
etac STL4E_gen 1,
auto_tac (temp_css addsimps2 Init_simps addSIs2 [PrimeDmd])
]);
qed_goalw "ensures" TLA.thy [leadsto_def]
"[| |- $P & N --> P` | Q`; \
\ |- ($P & N) & A --> Q` \
\ |] ==> |- []N & []([]P --> <>A) --> (P ~> Q)"
(fn [p1,p2] => [Clarsimp_tac 1,
etac STL4Edup 1, atac 1,
Clarsimp_tac 1,
subgoal_tac "sigmaa |= []($P --> P` | Q`)" 1,
dtac (temp_use unless) 1,
clarsimp_tac (temp_css addSDs2 [INV1]) 1,
rtac ((temp_use (p2 RS DmdImpl)) RS (temp_use DmdPrime)) 1,
force_tac (temp_css addSIs2 [BoxDmd_simple]
addsimps2 [split_box_conj,box_stp_act]) 1,
force_tac (temp_css addEs2 [STL4E] addDs2 [p1]) 1
]);
qed_goal "ensures_simple" TLA.thy
"[| |- $P & N --> P` | Q`; \
\ |- ($P & N) & A --> Q` \
\ |] ==> |- []N & []<>A --> (P ~> Q)"
(fn prems => [Clarsimp_tac 1,
rtac (temp_use ensures) 1,
TRYALL (ares_tac prems),
force_tac (temp_css addSEs2 [STL4E]) 1
]);
qed_goal "EnsuresInfinite" TLA.thy
"[| sigma |= []<>P; sigma |= []A; |- A & $P --> Q` |] ==> sigma |= []<>Q"
(fn prems => [REPEAT (resolve_tac (prems @ [temp_use leadsto_infinite,
temp_use EnsuresLeadsto]) 1)]);
(*** Gronning's lattice rules (taken from TLP) ***)
section "Lattice rules";
qed_goalw "LatticeReflexivity" TLA.thy [leadsto_def] "|- F ~> F"
(fn _ => [REPEAT (resolve_tac [necT,InitDmd_gen] 1)]);
qed_goalw "LatticeTransitivity" TLA.thy [leadsto_def]
"|- (G ~> H) & (F ~> G) --> (F ~> H)"
(fn _ => [Clarsimp_tac 1,
etac dup_boxE 1, (* [][](Init G --> H) *)
merge_box_tac 1,
clarsimp_tac (temp_css addSEs2 [STL4E]) 1,
rtac dup_dmdD 1,
subgoal_tac "sigmaa |= <>Init G" 1,
etac DmdImpl2 1, atac 1,
asm_simp_tac (simpset() addsimps [dmdInitD]) 1
]);
qed_goalw "LatticeDisjunctionElim1" TLA.thy [leadsto_def]
"|- (F | G ~> H) --> (F ~> H)"
(fn _ => [ auto_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E]) ]);
qed_goalw "LatticeDisjunctionElim2" TLA.thy [leadsto_def]
"|- (F | G ~> H) --> (G ~> H)"
(fn _ => [ auto_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E]) ]);
qed_goalw "LatticeDisjunctionIntro" TLA.thy [leadsto_def]
"|- (F ~> H) & (G ~> H) --> (F | G ~> H)"
(fn _ => [Clarsimp_tac 1,
merge_box_tac 1,
auto_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E])
]);
qed_goal "LatticeDisjunction" TLA.thy
"|- (F | G ~> H) = ((F ~> H) & (G ~> H))"
(fn _ => [auto_tac (temp_css addIs2 [LatticeDisjunctionIntro,
LatticeDisjunctionElim1, LatticeDisjunctionElim2])]);
qed_goal "LatticeDiamond" TLA.thy
"|- (A ~> B | C) & (B ~> D) & (C ~> D) --> (A ~> D)"
(fn _ => [Clarsimp_tac 1,
subgoal_tac "sigma |= (B | C) ~> D" 1,
eres_inst_tac [("G", "LIFT (B | C)")] (temp_use LatticeTransitivity) 1,
ALLGOALS (fast_tac (temp_cs addSIs [LatticeDisjunctionIntro]))
]);
qed_goal "LatticeTriangle" TLA.thy
"|- (A ~> D | B) & (B ~> D) --> (A ~> D)"
(fn _ => [Clarsimp_tac 1,
subgoal_tac "sigma |= (D | B) ~> D" 1,
eres_inst_tac [("G", "LIFT (D | B)")] (temp_use LatticeTransitivity) 1, atac 1,
auto_tac (temp_css addSIs2 [LatticeDisjunctionIntro]
addIs2 [LatticeReflexivity])
]);
qed_goal "LatticeTriangle2" TLA.thy
"|- (A ~> B | D) & (B ~> D) --> (A ~> D)"
(fn _ => [Clarsimp_tac 1,
subgoal_tac "sigma |= B | D ~> D" 1,
eres_inst_tac [("G", "LIFT (B | D)")] (temp_use LatticeTransitivity) 1, atac 1,
auto_tac (temp_css addSIs2 [LatticeDisjunctionIntro]
addIs2 [LatticeReflexivity])
]);
(*** Lamport's fairness rules ***)
section "Fairness rules";
qed_goal "WF1" TLA.thy
"[| |- $P & N --> P` | Q`; \
\ |- ($P & N) & <A>_v --> Q`; \
\ |- $P & N --> $(Enabled(<A>_v)) |] \
\ ==> |- []N & WF(A)_v --> (P ~> Q)" (fn prems => [
clarsimp_tac (temp_css addSDs2 [BoxWFI]) 1,
rtac (temp_use ensures) 1,
TRYALL (ares_tac prems),
etac STL4Edup 1, atac 1,
clarsimp_tac (temp_css addsimps2 [WF_def]) 1,
rtac (temp_use STL2) 1,
clarsimp_tac (temp_css addSEs2 [mp] addSIs2 [InitDmd]) 1,
resolve_tac ((map temp_use (prems RL [STL4])) RL [box_stp_actD]) 1,
asm_simp_tac (simpset() addsimps [split_box_conj,box_stp_actI]) 1
]);
(* Sometimes easier to use; designed for action B rather than state predicate Q *)
qed_goalw "WF_leadsto" TLA.thy [leadsto_def]
"[| |- N & $P --> $Enabled (<A>_v); \
\ |- N & <A>_v --> B; \
\ |- [](N & [~A]_v) --> stable P |] \
\ ==> |- []N & WF(A)_v --> (P ~> B)"
(fn [prem1,prem2,prem3]
=> [clarsimp_tac (temp_css addSDs2 [BoxWFI]) 1,
etac STL4Edup 1, atac 1,
Clarsimp_tac 1,
rtac (temp_use (prem2 RS DmdImpl)) 1,
rtac (temp_use BoxDmd_simple) 1, atac 1,
rtac classical 1,
rtac (temp_use STL2) 1,
clarsimp_tac (temp_css addsimps2 [WF_def] addSEs2 [mp] addSIs2 [InitDmd]) 1,
rtac ((temp_use (prem1 RS STL4)) RS box_stp_actD) 1,
asm_simp_tac (simpset() addsimps [split_box_conj,box_stp_act]) 1,
etac (temp_use INV1) 1,
rtac (temp_use prem3) 1,
asm_full_simp_tac (simpset() addsimps [split_box_conj,temp_use NotDmd,not_angle]) 1
]);
qed_goal "SF1" TLA.thy
"[| |- $P & N --> P` | Q`; \
\ |- ($P & N) & <A>_v --> Q`; \
\ |- []P & []N & []F --> <>Enabled(<A>_v) |] \
\ ==> |- []N & SF(A)_v & []F --> (P ~> Q)"
(fn prems => [
clarsimp_tac (temp_css addSDs2 [BoxSFI]) 1,
rtac (temp_use ensures) 1,
TRYALL (ares_tac prems),
eres_inst_tac [("F","F")] dup_boxE 1,
merge_temp_box_tac 1,
etac STL4Edup 1, atac 1,
clarsimp_tac (temp_css addsimps2 [SF_def]) 1,
rtac (temp_use STL2) 1, etac mp 1,
resolve_tac (map temp_use (prems RL [STL4])) 1,
asm_simp_tac (simpset() addsimps [split_box_conj, STL3]) 1
]);
qed_goal "WF2" TLA.thy
"[| |- N & <B>_f --> <M>_g; \
\ |- $P & P` & <N & A>_f --> B; \
\ |- P & Enabled(<M>_g) --> Enabled(<A>_f); \
\ |- [](N & [~B]_f) & WF(A)_f & []F & <>[]Enabled(<M>_g) --> <>[]P |] \
\ ==> |- []N & WF(A)_f & []F --> WF(M)_g"
(fn [prem1,prem2,prem3,prem4] => [
clarsimp_tac (temp_css addSDs2 [BoxWFI, (temp_use BoxDmdBox) RS iffD2]
addsimps2 [read_instantiate [("A","M")] WF_def]) 1,
eres_inst_tac [("F","F")] dup_boxE 1,
merge_temp_box_tac 1,
etac STL4Edup 1, atac 1,
clarsimp_tac (temp_css addSIs2 [(temp_use BoxDmd_simple) RS (temp_use (prem1 RS DmdImpl))]) 1,
rtac classical 1,
subgoal_tac "sigmaa |= <>(($P & P` & N) & <A>_f)" 1,
force_tac (temp_css addsimps2 [angle_def] addSIs2 [prem2] addSEs2 [DmdImplE]) 1,
rtac (temp_use (rewrite_rule [temp_rewrite DmdDmd] (BoxDmd_simple RS DmdImpl))) 1,
asm_full_simp_tac (simpset() addsimps [temp_use NotDmd, not_angle]) 1,
merge_act_box_tac 1,
forward_tac [temp_use prem4] 1, TRYALL atac,
dtac (temp_use STL6) 1, atac 1,
eres_inst_tac [("V","sigmaa |= <>[]P")] thin_rl 1,
eres_inst_tac [("V","sigmaa |= []F")] thin_rl 1,
dtac (temp_use BoxWFI) 1,
eres_inst_tac [("F", "ACT N & [~B]_f")] dup_boxE 1,
merge_temp_box_tac 1,
etac DmdImpldup 1, atac 1,
auto_tac (temp_css addsimps2 [split_box_conj,STL3,WF_Box,box_stp_act]),
force_tac (temp_css addSEs2 [read_instantiate [("P","P")] TLA2E]) 1,
rtac (temp_use STL2) 1,
force_tac (temp_css addsimps2 [WF_def,split_box_conj] addSEs2 [mp]
addSIs2 [InitDmd, prem3 RS STL4]) 1
]);
qed_goal "SF2" TLA.thy
"[| |- N & <B>_f --> <M>_g; \
\ |- $P & P` & <N & A>_f --> B; \
\ |- P & Enabled(<M>_g) --> Enabled(<A>_f); \
\ |- [](N & [~B]_f) & SF(A)_f & []F & []<>Enabled(<M>_g) --> <>[]P |] \
\ ==> |- []N & SF(A)_f & []F --> SF(M)_g"
(fn [prem1,prem2,prem3,prem4] => [
clarsimp_tac (temp_css addSDs2 [BoxSFI]
addsimps2 [read_instantiate [("A","M")] SF_def]) 1,
eres_inst_tac [("F","F")] dup_boxE 1,
eres_inst_tac [("F","TEMP <>Enabled(<M>_g)")] dup_boxE 1,
merge_temp_box_tac 1,
etac STL4Edup 1, atac 1,
clarsimp_tac (temp_css addSIs2 [(temp_use BoxDmd_simple) RS (temp_use (prem1 RS DmdImpl))]) 1,
rtac classical 1,
subgoal_tac "sigmaa |= <>(($P & P` & N) & <A>_f)" 1,
force_tac (temp_css addsimps2 [angle_def] addSIs2 [prem2] addSEs2 [DmdImplE]) 1,
rtac (temp_use (rewrite_rule [temp_rewrite DmdDmd] (BoxDmd_simple RS DmdImpl))) 1,
asm_full_simp_tac (simpset() addsimps [temp_use NotDmd, not_angle]) 1,
merge_act_box_tac 1,
forward_tac [temp_use prem4] 1, TRYALL atac,
eres_inst_tac [("V","sigmaa |= []F")] thin_rl 1,
dtac (temp_use BoxSFI) 1,
eres_inst_tac [("F","TEMP <>Enabled(<M>_g)")] dup_boxE 1,
eres_inst_tac [("F", "ACT N & [~B]_f")] dup_boxE 1,
merge_temp_box_tac 1,
etac DmdImpldup 1, atac 1,
auto_tac (temp_css addsimps2 [split_box_conj,STL3,SF_Box,box_stp_act]),
force_tac (temp_css addSEs2 [read_instantiate [("P","P")] TLA2E]) 1,
rtac (temp_use STL2) 1,
force_tac (temp_css addsimps2 [SF_def,split_box_conj] addSEs2 [mp,InfImpl]
addSIs2 [prem3]) 1
]);
(* ------------------------------------------------------------------------- *)
(*** Liveness proofs by well-founded orderings ***)
(* ------------------------------------------------------------------------- *)
section "Well-founded orderings";
qed_goal "wf_leadsto" TLA.thy
"[| wf r; \
\ !!x. sigma |= F x ~> (G | (? y. #((y,x):r) & F y)) \
\ |] ==> sigma |= F x ~> G"
(fn p1::prems =>
[rtac (p1 RS wf_induct) 1,
rtac (temp_use LatticeTriangle) 1,
resolve_tac prems 1,
auto_tac (temp_css addsimps2 [leadsto_exists]),
case_tac "(y,x):r" 1,
Force_tac 1,
force_tac (temp_css addsimps2 leadsto_def::Init_simps addSIs2 [necT]) 1]);
(* If r is well-founded, state function v cannot decrease forever *)
qed_goal "wf_not_box_decrease" TLA.thy
"!!r. wf r ==> |- [][ (v`, $v) : #r ]_v --> <>[][#False]_v"
(fn _ => [Clarsimp_tac 1,
rtac ccontr 1,
subgoal_tac "sigma |= (? x. v=#x) ~> #False" 1,
dtac ((temp_use leadsto_false) RS iffD1 RS (temp_use STL2_gen)) 1,
force_tac (temp_css addsimps2 Init_defs) 1,
clarsimp_tac (temp_css addsimps2 [leadsto_exists,not_square]@more_temp_simps) 1,
etac wf_leadsto 1,
rtac (temp_use ensures_simple) 1, TRYALL atac,
auto_tac (temp_css addsimps2 [square_def,angle_def])
]);
(* "wf r ==> |- <>[][ (v`, $v) : #r ]_v --> <>[][#False]_v" *)
bind_thm("wf_not_dmd_box_decrease",
standard(rewrite_rule more_temp_simps (wf_not_box_decrease RS DmdImpl)));
(* If there are infinitely many steps where v decreases, then there
have to be infinitely many non-stuttering steps where v doesn't decrease.
*)
qed_goal "wf_box_dmd_decrease" TLA.thy
"wf r ==> |- []<>((v`, $v) : #r) --> []<><(v`, $v) ~: #r>_v"
(fn [prem] => [
Clarsimp_tac 1,
rtac ccontr 1,
asm_full_simp_tac (simpset() addsimps not_angle::more_temp_simps) 1,
dtac (prem RS (temp_use wf_not_dmd_box_decrease)) 1,
dtac (temp_use BoxDmdDmdBox) 1, atac 1,
subgoal_tac "sigma |= []<>((#False)::action)" 1,
Force_tac 1,
etac STL4E 1,
rtac DmdImpl 1,
force_tac (temp_css addIs2 [prem RS wf_irrefl]) 1
]);
(* In particular, for natural numbers, if n decreases infinitely often
then it has to increase infinitely often.
*)
qed_goal "nat_box_dmd_decrease" TLA.thy
"!!n::nat stfun. |- []<>(n` < $n) --> []<>($n < n`)"
(K [Clarsimp_tac 1,
subgoal_tac "sigma |= []<><~( (n`,$n) : #less_than )>_n" 1,
etac thin_rl 1, etac STL4E 1, rtac DmdImpl 1,
clarsimp_tac (temp_css addsimps2 [angle_def]) 1,
rtac nat_less_cases 1,
Auto_tac,
rtac (temp_use wf_box_dmd_decrease) 1,
auto_tac (temp_css addSEs2 [STL4E,DmdImplE])
]);
(* ------------------------------------------------------------------------- *)
(*** Flexible quantification over state variables ***)
(* ------------------------------------------------------------------------- *)
section "Flexible quantification";
qed_goal "aallI" TLA.thy
"[| basevars vs; (!!x. basevars (x,vs) ==> sigma |= F x) |] ==> sigma |= (AALL x. F x)"
(fn [prem1,prem2] => [auto_tac (temp_css addsimps2 [aall_def] addSEs2 [eexE]
addSIs2 [prem1] addSDs2 [prem2])]);
qed_goalw "aallE" TLA.thy [aall_def] "|- (AALL x. F x) --> F x"
(K [Clarsimp_tac 1, etac swap 1,
force_tac (temp_css addSIs2 [eexI]) 1]);
(* monotonicity of quantification *)
qed_goal "eex_mono" TLA.thy
"[| sigma |= EEX x. F x; !!x. sigma |= F x --> G x |] ==> sigma |= EEX x. G x"
(fn [min,maj] => [rtac (unit_base RS (min RS eexE)) 1,
rtac (temp_use eexI) 1,
etac ((rewrite_rule intensional_rews maj) RS mp) 1
]);
qed_goal "aall_mono" TLA.thy
"[| sigma |= AALL x. F(x); !!x. sigma |= F(x) --> G(x) |] ==> sigma |= AALL x. G(x)"
(fn [min,maj] => [rtac (unit_base RS aallI) 1,
rtac ((rewrite_rule intensional_rews maj) RS mp) 1,
rtac (min RS (temp_use aallE)) 1
]);
(* Derived history introduction rule *)
qed_goal "historyI" TLA.thy
"[| sigma |= Init I; sigma |= []N; basevars vs; \
\ (!!h. basevars(h,vs) ==> |- I & h = ha --> HI h); \
\ (!!h s t. [| basevars(h,vs); N (s,t); h t = hb (h s) (s,t) |] ==> HN h (s,t)) \
\ |] ==> sigma |= EEX h. Init (HI h) & [](HN h)"
(fn [p1,p2,p3,p4,p5]
=> [rtac ((temp_use history) RS eexE) 1,
rtac p3 1,
rtac (temp_use eexI) 1,
Clarsimp_tac 1, rtac conjI 1,
cut_facts_tac [p2] 2,
merge_box_tac 2,
force_tac (temp_css addSEs2 [STL4E,p5]) 2,
cut_facts_tac [p1] 1,
force_tac (temp_css addsimps2 Init_defs addSEs2 [p4]) 1
]);
(* ----------------------------------------------------------------------
example of a history variable: existence of a clock
Goal "|- EEX h. Init(h = #True) & [](h` = (~$h))";
by (rtac tempI 1);
by (rtac historyI 1);
by (REPEAT (force_tac (temp_css addsimps2 Init_defs addIs2 [unit_base, necT]) 1));
(** solved **)
---------------------------------------------------------------------- *)