(* File: TLA/TLA.ML ID: $Id: TLA.ML,v 1.16 2005/09/07 18:22:40 wenzelm Exp $ Author: Stephan Merz Copyright: 1998 University of Munich Lemmas and tactics for temporal reasoning. *) (* Specialize intensional introduction/elimination rules for temporal formulas *) val [prem] = goal (the_context ()) "(!!sigma. sigma |= (F::temporal)) ==> |- F"; by (REPEAT (resolve_tac [prem,intI] 1)); qed "tempI"; val [prem] = goal (the_context ()) "|- (F::temporal) ==> sigma |= F"; by (rtac (prem RS intD) 1); qed "tempD"; (* ======== 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)); Goalw [dmd_def] "TEMP <>F == TEMP <> Init F"; by (rewtac (read_instantiate [("F", "LIFT ~F")] boxInit)); by (simp_tac (simpset() addsimps Init_simps) 1); qed "dmdInit"; 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 *) Goal "|- []F --> Init F"; by (rewtac (read_instantiate [("F", "F")] boxInit)); by (rtac STL2 1); qed "STL2_gen"; (* see also STL2_pr below: "|- []P --> Init P & Init (P`)" *) (* Dual versions for <> *) Goalw [dmd_def] "|- F --> <> F"; by (auto_tac (temp_css addSDs2 [STL2])); qed "InitDmd"; Goal "|- Init F --> <>F"; by (Clarsimp_tac 1); by (dtac (temp_use InitDmd) 1); by (asm_full_simp_tac (simpset() addsimps [dmdInitD]) 1); qed "InitDmd_gen"; (* ------------------------ STL3 ------------------------------------------- *) Goal "|- ([][]F) = ([]F)"; by (force_tac (temp_css addEs2 [transT,STL2]) 1); qed "STL3"; (* 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 <> *) Goal "|- (<><>F) = (<>F)"; by (auto_tac (temp_css addsimps2 [dmd_def,STL3])); qed "DmdDmd"; bind_thm("dup_dmdE", make_elim((temp_unlift DmdDmd) RS iffD2)); bind_thm("dup_dmdD", (temp_unlift DmdDmd) RS iffD1); (* ------------------------ STL4 ------------------------------------------- *) val [prem] = goal (the_context ()) "|- F --> G ==> |- []F --> []G"; by (Clarsimp_tac 1); by (rtac (temp_use normalT) 1); by (rtac (temp_use (prem RS necT)) 1); by (atac 1); qed "STL4"; (* Unlifted version as an elimination rule *) val prems = goal (the_context ()) "[| sigma |= []F; |- F --> G |] ==> sigma |= []G"; by (REPEAT (resolve_tac (prems @ [temp_use STL4]) 1)); qed "STL4E"; val [prem] = goal (the_context ()) "|- Init F --> Init G ==> |- []F --> []G"; by (rtac (rewrite_rule [boxInitD] (prem RS STL4)) 1); qed "STL4_gen"; val prems = goal (the_context ()) "[| sigma |= []F; |- Init F --> Init G |] ==> sigma |= []G"; by (REPEAT (resolve_tac (prems @ [temp_use STL4_gen]) 1)); qed "STL4E_gen"; (* see also STL4Edup below, which allows an auxiliary boxed formula: []A /\ F => G ----------------- []A /\ []F => []G *) (* The dual versions for <> *) val [prem] = goalw (the_context ()) [dmd_def] "|- F --> G ==> |- <>F --> <>G"; by (fast_tac (temp_cs addSIs [prem] addSEs [STL4E]) 1); qed "DmdImpl"; val prems = goal (the_context ()) "[| sigma |= <>F; |- F --> G |] ==> sigma |= <>G"; by (REPEAT (resolve_tac (prems @ [temp_use DmdImpl]) 1)); qed "DmdImplE"; (* ------------------------ STL5 ------------------------------------------- *) Goal "|- ([]F & []G) = ([](F & G))"; by Auto_tac; by (subgoal_tac "sigma |= [](G --> (F & G))" 1); by (etac (temp_use normalT) 1); by (ALLGOALS (fast_tac (temp_cs addSEs [STL4E]))); qed "STL5"; (* 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! *) val prems = goal (the_context ()) "[| sigma |= []F; sigma |= []G; sigma |= [](F&G) ==> PROP R |] ==> PROP R"; by (REPEAT (resolve_tac (prems @ [(temp_unlift STL5) RS iffD1, conjI]) 1)); qed "box_conjE"; (* 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)); Goal "|- (<>(F | G)) = (<>F | <>G)"; by (auto_tac (temp_css addsimps2 [dmd_def,split_box_conj])); by (ALLGOALS (EVERY' [etac contrapos_np, merge_box_tac, fast_tac (temp_cs addSEs [STL4E])])); qed "DmdOr"; Goal "|- (EX x. <>(F x)) = (<>(EX x. F x))"; by (auto_tac (temp_css addsimps2 [dmd_def,Not_Rex,all_box])); qed "exT"; bind_thm("ex_dmd", standard((temp_unlift exT) RS sym)); Goal "!!sigma. [| sigma |= []A; sigma |= []F; |- F & []A --> G |] ==> sigma |= []G"; by (etac dup_boxE 1); by (merge_box_tac 1); by (etac STL4E 1); by (atac 1); qed "STL4Edup"; Goalw [dmd_def] "!!sigma. [| sigma |= <>F; sigma |= [](F --> G) |] ==> sigma |= <>G"; by Auto_tac; by (etac notE 1); by (merge_box_tac 1); by (fast_tac (temp_cs addSEs [STL4E]) 1); qed "DmdImpl2"; val [prem1,prem2,prem3] = goal (the_context ()) "[| sigma |= []<>F; sigma |= []G; |- F & G --> H |] ==> sigma |= []<>H"; by (cut_facts_tac [prem1,prem2] 1); by (eres_inst_tac [("F","G")] dup_boxE 1); by (merge_box_tac 1); by (fast_tac (temp_cs addSEs [STL4E,DmdImpl2] addSIs [prem3]) 1); qed "InfImpl"; (* ------------------------ STL6 ------------------------------------------- *) (* Used in the proof of STL6, but useful in itself. *) Goalw [dmd_def] "|- []F & <>G --> <>([]F & G)"; by (Clarsimp_tac 1); by (etac dup_boxE 1); by (merge_box_tac 1); by (etac contrapos_np 1); by (fast_tac (temp_cs addSEs [STL4E]) 1); qed "BoxDmd"; (* weaker than BoxDmd, but more polymorphic (and often just right) *) Goalw [dmd_def] "|- []F & <>G --> <>(F & G)"; by (Clarsimp_tac 1); by (merge_box_tac 1); by (fast_tac (temp_cs addSEs [notE,STL4E]) 1); qed "BoxDmd_simple"; Goalw [dmd_def] "|- []F & <>G --> <>(G & F)"; by (Clarsimp_tac 1); by (merge_box_tac 1); by (fast_tac (temp_cs addSEs [notE,STL4E]) 1); qed "BoxDmd2_simple"; val [p1,p2,p3] = goal (the_context ()) "[| sigma |= []A; sigma |= <>F; |- []A & F --> G |] ==> sigma |= <>G"; by (rtac ((p2 RS (p1 RS (temp_use BoxDmd))) RS DmdImplE) 1); by (rtac p3 1); qed "DmdImpldup"; Goal "|- <>[]F & <>[]G --> <>[](F & G)"; by (auto_tac (temp_css addsimps2 [symmetric (temp_rewrite STL5)])); by (dtac (temp_use linT) 1); by (atac 1); by (etac thin_rl 1); by (rtac ((temp_unlift DmdDmd) RS iffD1) 1); by (etac disjE 1); by (etac DmdImplE 1); by (rtac BoxDmd 1); by (etac DmdImplE 1); by Auto_tac; by (dtac (temp_use BoxDmd) 1); by (atac 1); by (etac thin_rl 1); by (fast_tac (temp_cs addSEs [DmdImplE]) 1); qed "STL6"; (* ------------------------ True / False ----------------------------------------- *) section "Simplification of constants"; Goal "|- ([]#P) = #P"; by (rtac tempI 1); by (case_tac "P" 1); by (auto_tac (temp_css addSIs2 [necT] addDs2 [STL2_gen] addsimps2 Init_simps)); qed "BoxConst"; Goalw [dmd_def] "|- (<>#P) = #P"; by (case_tac "P" 1); by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [BoxConst]))); qed "DmdConst"; 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"; Goalw [dmd_def] "|- (~[]F) = (<>~F)"; by (Simp_tac 1); qed "NotBox"; Goalw [dmd_def] "|- (~<>F) = ([]~F)"; by (Simp_tac 1); qed "NotDmd"; (* 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]); Goal "|- ([]<>[]F) = (<>[]F)"; by (auto_tac (temp_css addSDs2 [STL2])); by (rtac ccontr 1); by (subgoal_tac "sigma |= <>[][]F & <>[]~[]F" 1); by (etac thin_rl 1); by Auto_tac; by (dtac (temp_use STL6) 1); by (atac 1); by (Asm_full_simp_tac 1); by (ALLGOALS (asm_full_simp_tac (simpset() addsimps more_temp_simps))); qed "BoxDmdBox"; Goalw [dmd_def] "|- (<>[]<>F) = ([]<>F)"; by (auto_tac (temp_css addsimps2 [rewrite_rule [dmd_def] BoxDmdBox])); qed "DmdBoxDmd"; val more_temp_simps = more_temp_simps @ (map temp_rewrite [BoxDmdBox, DmdBoxDmd]); (* ------------------------ Miscellaneous ----------------------------------- *) Goal "!!sigma. [| sigma |= []F | []G |] ==> sigma |= [](F | G)"; by (fast_tac (temp_cs addSEs [STL4E]) 1); qed "BoxOr"; (* "persistently implies infinitely often" *) Goal "|- <>[]F --> []<>F"; by (Clarsimp_tac 1); by (rtac ccontr 1); by (asm_full_simp_tac (simpset() addsimps more_temp_simps) 1); by (dtac (temp_use STL6) 1); by (atac 1); by (Asm_full_simp_tac 1); qed "DBImplBD"; Goal "|- []<>F & <>[]G --> []<>(F & G)"; by (Clarsimp_tac 1); by (rtac ccontr 1); by (rewrite_goals_tac more_temp_simps); by (dtac (temp_use STL6) 1); by (atac 1); by (subgoal_tac "sigma |= <>[]~F" 1); by (force_tac (temp_css addsimps2 [dmd_def]) 1); by (fast_tac (temp_cs addEs [DmdImplE,STL4E]) 1); qed "BoxDmdDmdBox"; (* ------------------------------------------------------------------------- *) (*** TLA-specific theorems: primed formulas ***) (* ------------------------------------------------------------------------- *) section "priming"; (* ------------------------ TLA2 ------------------------------------------- *) Goal "|- []P --> Init P & Init P`"; by (fast_tac (temp_cs addSIs [primeI, STL2_gen]) 1); qed "STL2_pr"; (* Auxiliary lemma allows priming of boxed actions *) Goal "|- []P --> []($P & P$)"; by (Clarsimp_tac 1); by (etac dup_boxE 1); by (rewtac boxInit_act); by (etac STL4E 1); by (auto_tac (temp_css addsimps2 Init_simps addSDs2 [STL2_pr])); qed "BoxPrime"; val prems = goal (the_context ()) "|- $P & P$ --> A ==> |- []P --> []A"; by (Clarsimp_tac 1); by (dtac (temp_use BoxPrime) 1); by (auto_tac (temp_css addsimps2 [Init_stp_act_rev] addSIs2 prems addSEs2 [STL4E])); qed "TLA2"; val prems = goal (the_context ()) "[| sigma |= []P; |- $P & P$ --> A |] ==> sigma |= []A"; by (REPEAT (resolve_tac (prems @ (prems RL [temp_use TLA2])) 1)); qed "TLA2E"; Goalw [dmd_def] "|- (<>P`) --> (<>P)"; by (fast_tac (temp_cs addSEs [TLA2E]) 1); qed "DmdPrime"; bind_thm("PrimeDmd", (temp_use InitDmd_gen) RS (temp_use DmdPrime)); (* ------------------------ INV1, stable --------------------------------------- *) section "stable, invariant"; val prems = goal (the_context ()) "[| sigma |= []H; sigma |= Init P; |- H --> (Init P & ~[]F --> Init(P`) & F) |] \ \ ==> sigma |= []F"; by (rtac (temp_use indT) 1); by (REPEAT (resolve_tac (prems @ (prems RL [STL4E])) 1)); qed "ind_rule"; Goalw [boxInit_act] "|- ([]$P) = ([]P)"; by (simp_tac (simpset() addsimps Init_simps) 1); qed "box_stp_act"; 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; Goalw [stable_def,boxInit_stp,boxInit_act] "|- (Init P) --> (stable P) --> []P"; by (Clarsimp_tac 1); by (etac ind_rule 1); by (auto_tac (temp_css addsimps2 Init_simps addEs2 [ind_rule])); qed "INV1"; Goalw [stable_def] "!!P. |- $P & A --> P` ==> |- []A --> stable P"; by (fast_tac (temp_cs addSEs [STL4E]) 1); qed "StableT"; val prems = goal (the_context ()) "[| sigma |= []A; |- $P & A --> P` |] ==> sigma |= stable P"; by (REPEAT (resolve_tac (prems @ [temp_use StableT]) 1)); qed "Stable"; (* Generalization of INV1 *) Goalw [stable_def] "|- (stable P) --> [](Init P --> []P)"; by (Clarsimp_tac 1); by (etac dup_boxE 1); by (force_tac (temp_css addsimps2 [stable_def] addEs2 [STL4E, INV1]) 1); qed "StableBox"; Goal "|- (stable P) & <>P --> <>[]P"; by (Clarsimp_tac 1); by (rtac DmdImpl2 1); by (etac (temp_use StableBox) 2); by (asm_simp_tac (simpset() addsimps [dmdInitD]) 1); qed "DmdStable"; (* ---------------- (Semi-)automatic invariant tactics ---------------------- *) (* inv_tac reduces goals of the form ... ==> sigma |= []P *) fun inv_tac css = SELECT_GOAL (EVERY [auto_tac css, TRY (merge_box_tac 1), rtac (temp_use INV1) 1, (* fail if the goal is not a box *) TRYALL (etac Stable)]); (* auto_inv_tac applies inv_tac and then tries to attack the subgoals; in simple cases it may be able to handle goals like |- MyProg --> []Inv. In these simple cases the simplifier seems to be more useful than the auto-tactic, which applies too much propositional logic and simplifies too late. *) fun auto_inv_tac ss = SELECT_GOAL ((inv_tac (claset(),ss) 1) THEN (TRYALL (action_simp_tac (ss addsimps [Init_stp,Init_act]) [] [squareE]))); Goalw [dmd_def] "|- []($P --> P` | Q`) --> (stable P) | <>Q"; by (clarsimp_tac (temp_css addSDs2 [BoxPrime]) 1); by (merge_box_tac 1); by (etac contrapos_np 1); by (fast_tac (temp_cs addSEs [Stable]) 1); qed "unless"; (* --------------------- Recursive expansions --------------------------------------- *) section "recursive expansions"; (* Recursive expansions of [] and <> for state predicates *) Goal "|- ([]P) = (Init P & []P`)"; by (auto_tac (temp_css addSIs2 [STL2_gen])); by (fast_tac (temp_cs addSEs [TLA2E]) 1); by (auto_tac (temp_css addsimps2 [stable_def] addSEs2 [INV1,STL4E])); qed "BoxRec"; Goalw [dmd_def, temp_rewrite BoxRec] "|- (<>P) = (Init P | <>P`)"; by (auto_tac (temp_css addsimps2 Init_simps)); qed "DmdRec"; Goal "!!sigma. [| sigma |= <>P; sigma |= []~P` |] ==> sigma |= Init P"; by (force_tac (temp_css addsimps2 [DmdRec,dmd_def]) 1); qed "DmdRec2"; Goal "|- ([]<>P) = ([]<>P`)"; by Auto_tac; by (rtac classical 1); by (rtac (temp_use DBImplBD) 1); by (subgoal_tac "sigma |= <>[]P" 1); by (fast_tac (temp_cs addSEs [DmdImplE,TLA2E]) 1); by (subgoal_tac "sigma |= <>[](<>P & []~P`)" 1); by (force_tac (temp_css addsimps2 [boxInit_stp] addSEs2 [DmdImplE,STL4E,DmdRec2]) 1); by (force_tac (temp_css addSIs2 [STL6] addsimps2 more_temp_simps) 1); by (fast_tac (temp_cs addIs [DmdPrime] addSEs [STL4E]) 1); qed "InfinitePrime"; val prems = goalw (the_context ()) [temp_rewrite InfinitePrime] "[| sigma |= []N; sigma |= []<>A; |- A & N --> P` |] ==> sigma |= []<>P"; by (rtac InfImpl 1); by (REPEAT (resolve_tac prems 1)); qed "InfiniteEnsures"; (* ------------------------ fairness ------------------------------------------- *) section "fairness"; (* alternative definitions of fairness *) Goalw [WF_def,dmd_def] "|- WF(A)_v = ([]<>~Enabled(<A>_v) | []<><A>_v)"; by (fast_tac temp_cs 1); qed "WF_alt"; Goalw [SF_def,dmd_def] "|- SF(A)_v = (<>[]~Enabled(<A>_v) | []<><A>_v)"; by (fast_tac temp_cs 1); qed "SF_alt"; (* theorems to "box" fairness conditions *) Goal "|- WF(A)_v --> []WF(A)_v"; by (auto_tac (temp_css addsimps2 (WF_alt::more_temp_simps) addSIs2 [BoxOr])); qed "BoxWFI"; Goal "|- ([]WF(A)_v) = WF(A)_v"; by (fast_tac (temp_cs addSIs [BoxWFI] addSDs [STL2]) 1); qed "WF_Box"; Goal "|- SF(A)_v --> []SF(A)_v"; by (auto_tac (temp_css addsimps2 (SF_alt::more_temp_simps) addSIs2 [BoxOr])); qed "BoxSFI"; Goal "|- ([]SF(A)_v) = SF(A)_v"; by (fast_tac (temp_cs addSIs [BoxSFI] addSDs [STL2]) 1); qed "SF_Box"; val more_temp_simps = more_temp_simps @ (map temp_rewrite [WF_Box, SF_Box]); Goalw [SF_def,WF_def] "|- SF(A)_v --> WF(A)_v"; by (fast_tac (temp_cs addSDs [DBImplBD]) 1); qed "SFImplWF"; (* 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 "~>"; Goalw [leadsto_def] "|- (Init F) & (F ~> G) --> <>G"; by (auto_tac (temp_css addSDs2 [STL2])); qed "leadsto_init"; (* |- F & (F ~> G) --> <>G *) bind_thm("leadsto_init_temp", rewrite_rule Init_simps (read_instantiate [("'a","behavior")] leadsto_init)); Goalw [leadsto_def] "|- ([]<>Init F --> []<>G) = (<>(F ~> G))"; by Auto_tac; by (asm_full_simp_tac (simpset() addsimps more_temp_simps) 1); by (fast_tac (temp_cs addSEs [DmdImplE,STL4E]) 1); by (fast_tac (temp_cs addSIs [InitDmd] addSEs [STL4E]) 1); by (subgoal_tac "sigma |= []<><>G" 1); by (asm_full_simp_tac (simpset() addsimps more_temp_simps) 1); by (dtac (temp_use BoxDmdDmdBox) 1); by (atac 1); by (fast_tac (temp_cs addSEs [DmdImplE,STL4E]) 1); qed "streett_leadsto"; Goal "|- []<>F & (F ~> G) --> []<>G"; by (Clarsimp_tac 1); by (etac ((temp_use InitDmd) RS ((temp_unlift streett_leadsto) RS iffD2 RS mp)) 1); by (asm_simp_tac (simpset() addsimps [dmdInitD]) 1); qed "leadsto_infinite"; (* In particular, strong fairness is a Streett condition. The following rules are sometimes easier to use than WF2 or SF2 below. *) Goalw [SF_def] "|- (Enabled(<A>_v) ~> <A>_v) --> SF(A)_v"; by (clarsimp_tac (temp_css addSEs2 [leadsto_infinite]) 1); qed "leadsto_SF"; Goal "|- (Enabled(<A>_v) ~> <A>_v) --> WF(A)_v"; by (clarsimp_tac (temp_css addSIs2 [SFImplWF, leadsto_SF]) 1); qed "leadsto_WF"; (* introduce an invariant into the proof of a leadsto assertion. []I --> ((P ~> Q) = (P /\ I ~> Q)) *) Goalw [leadsto_def] "|- []I & (P & I ~> Q) --> (P ~> Q)"; by (Clarsimp_tac 1); by (etac STL4Edup 1); by (atac 1); by (auto_tac (temp_css addsimps2 Init_simps addSDs2 [STL2_gen])); qed "INV_leadsto"; Goalw [leadsto_def,dmd_def] "|- (Init F & []~G ~> G) --> (F ~> G)"; by (force_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E]) 1); qed "leadsto_classical"; Goalw [leadsto_def] "|- (F ~> #False) = ([]~F)"; by (simp_tac (simpset() addsimps [boxNotInitD]) 1); qed "leadsto_false"; Goalw [leadsto_def] "|- ((EX x. F x) ~> G) = (ALL x. (F x ~> G))"; by (auto_tac (temp_css addsimps2 allT::Init_simps addSEs2 [STL4E])); qed "leadsto_exists"; (* basic leadsto properties, cf. Unity *) Goalw [leadsto_def] "|- [](Init F --> Init G) --> (F ~> G)"; by (auto_tac (temp_css addSIs2 [InitDmd_gen] addSEs2 [STL4E_gen] addsimps2 Init_simps)); qed "ImplLeadsto_gen"; bind_thm("ImplLeadsto", rewrite_rule Init_simps (read_instantiate [("'a","behavior"), ("'b","behavior")] ImplLeadsto_gen)); Goal "!!F G. |- F --> G ==> |- F ~> G"; by (auto_tac (temp_css addsimps2 [Init_def] addSIs2 [ImplLeadsto_gen,necT])); qed "ImplLeadsto_simple"; val [prem] = goalw (the_context ()) [leadsto_def] "|- A & $P --> Q` ==> |- []A --> (P ~> Q)"; by (clarsimp_tac (temp_css addSEs2 [INV_leadsto]) 1); by (etac STL4E_gen 1); by (auto_tac (temp_css addsimps2 Init_defs addSIs2 [PrimeDmd,prem])); qed "EnsuresLeadsto"; Goalw [leadsto_def] "|- []($P --> Q`) --> (P ~> Q)"; by (Clarsimp_tac 1); by (etac STL4E_gen 1); by (auto_tac (temp_css addsimps2 Init_simps addSIs2 [PrimeDmd])); qed "EnsuresLeadsto2"; val [p1,p2] = goalw (the_context ()) [leadsto_def] "[| |- $P & N --> P` | Q`; \ \ |- ($P & N) & A --> Q` \ \ |] ==> |- []N & []([]P --> <>A) --> (P ~> Q)"; by (Clarsimp_tac 1); by (etac STL4Edup 1); by (atac 1); by (Clarsimp_tac 1); by (subgoal_tac "sigmaa |= []($P --> P` | Q`)" 1); by (dtac (temp_use unless) 1); by (clarsimp_tac (temp_css addSDs2 [INV1]) 1); by (rtac ((temp_use (p2 RS DmdImpl)) RS (temp_use DmdPrime)) 1); by (force_tac (temp_css addSIs2 [BoxDmd_simple] addsimps2 [split_box_conj,box_stp_act]) 1); by (force_tac (temp_css addEs2 [STL4E] addDs2 [p1]) 1); qed "ensures"; val prems = goal (the_context ()) "[| |- $P & N --> P` | Q`; \ \ |- ($P & N) & A --> Q` \ \ |] ==> |- []N & []<>A --> (P ~> Q)"; by (Clarsimp_tac 1); by (rtac (temp_use ensures) 1); by (TRYALL (ares_tac prems)); by (force_tac (temp_css addSEs2 [STL4E]) 1); qed "ensures_simple"; val prems = goal (the_context ()) "[| sigma |= []<>P; sigma |= []A; |- A & $P --> Q` |] ==> sigma |= []<>Q"; by (REPEAT (resolve_tac (prems @ (map temp_use [leadsto_infinite, EnsuresLeadsto])) 1)); qed "EnsuresInfinite"; (*** Gronning's lattice rules (taken from TLP) ***) section "Lattice rules"; Goalw [leadsto_def] "|- F ~> F"; by (REPEAT (resolve_tac [necT,InitDmd_gen] 1)); qed "LatticeReflexivity"; Goalw [leadsto_def] "|- (G ~> H) & (F ~> G) --> (F ~> H)"; by (Clarsimp_tac 1); by (etac dup_boxE 1); (* [][](Init G --> H) *) by (merge_box_tac 1); by (clarsimp_tac (temp_css addSEs2 [STL4E]) 1); by (rtac dup_dmdD 1); by (subgoal_tac "sigmaa |= <>Init G" 1); by (etac DmdImpl2 1); by (atac 1); by (asm_simp_tac (simpset() addsimps [dmdInitD]) 1); qed "LatticeTransitivity"; Goalw [leadsto_def] "|- (F | G ~> H) --> (F ~> H)"; by (auto_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E])); qed "LatticeDisjunctionElim1"; Goalw [leadsto_def] "|- (F | G ~> H) --> (G ~> H)"; by (auto_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E])); qed "LatticeDisjunctionElim2"; Goalw [leadsto_def] "|- (F ~> H) & (G ~> H) --> (F | G ~> H)"; by (Clarsimp_tac 1); by (merge_box_tac 1); by (auto_tac (temp_css addsimps2 Init_simps addSEs2 [STL4E])); qed "LatticeDisjunctionIntro"; Goal "|- (F | G ~> H) = ((F ~> H) & (G ~> H))"; by (auto_tac (temp_css addIs2 [LatticeDisjunctionIntro, LatticeDisjunctionElim1, LatticeDisjunctionElim2])); qed "LatticeDisjunction"; Goal "|- (A ~> B | C) & (B ~> D) & (C ~> D) --> (A ~> D)"; by (Clarsimp_tac 1); by (subgoal_tac "sigma |= (B | C) ~> D" 1); by (eres_inst_tac [("G", "LIFT (B | C)")] (temp_use LatticeTransitivity) 1); by (ALLGOALS (fast_tac (temp_cs addSIs [LatticeDisjunctionIntro]))); qed "LatticeDiamond"; Goal "|- (A ~> D | B) & (B ~> D) --> (A ~> D)"; by (Clarsimp_tac 1); by (subgoal_tac "sigma |= (D | B) ~> D" 1); by (eres_inst_tac [("G", "LIFT (D | B)")] (temp_use LatticeTransitivity) 1); by (atac 1); by (auto_tac (temp_css addIs2 [LatticeDisjunctionIntro,LatticeReflexivity])); qed "LatticeTriangle"; Goal "|- (A ~> B | D) & (B ~> D) --> (A ~> D)"; by (Clarsimp_tac 1); by (subgoal_tac "sigma |= B | D ~> D" 1); by (eres_inst_tac [("G", "LIFT (B | D)")] (temp_use LatticeTransitivity) 1); by (atac 1); by (auto_tac (temp_css addIs2 [LatticeDisjunctionIntro,LatticeReflexivity])); qed "LatticeTriangle2"; (*** Lamport's fairness rules ***) section "Fairness rules"; val prems = goal (the_context ()) "[| |- $P & N --> P` | Q`; \ \ |- ($P & N) & <A>_v --> Q`; \ \ |- $P & N --> $(Enabled(<A>_v)) |] \ \ ==> |- []N & WF(A)_v --> (P ~> Q)"; by (clarsimp_tac (temp_css addSDs2 [BoxWFI]) 1); by (rtac (temp_use ensures) 1); by (TRYALL (ares_tac prems)); by (etac STL4Edup 1); by (atac 1); by (clarsimp_tac (temp_css addsimps2 [WF_def]) 1); by (rtac (temp_use STL2) 1); by (clarsimp_tac (temp_css addSEs2 [mp] addSIs2 [InitDmd]) 1); by (resolve_tac ((map temp_use (prems RL [STL4])) RL [box_stp_actD]) 1); by (asm_simp_tac (simpset() addsimps [split_box_conj,box_stp_actI]) 1); qed "WF1"; (* Sometimes easier to use; designed for action B rather than state predicate Q *) val [prem1,prem2,prem3] = goalw (the_context ()) [leadsto_def] "[| |- N & $P --> $Enabled (<A>_v); \ \ |- N & <A>_v --> B; \ \ |- [](N & [~A]_v) --> stable P |] \ \ ==> |- []N & WF(A)_v --> (P ~> B)"; by (clarsimp_tac (temp_css addSDs2 [BoxWFI]) 1); by (etac STL4Edup 1); by (atac 1); by (Clarsimp_tac 1); by (rtac (temp_use (prem2 RS DmdImpl)) 1); by (rtac (temp_use BoxDmd_simple) 1); by (atac 1); by (rtac classical 1); by (rtac (temp_use STL2) 1); by (clarsimp_tac (temp_css addsimps2 [WF_def] addSEs2 [mp] addSIs2 [InitDmd]) 1); by (rtac ((temp_use (prem1 RS STL4)) RS box_stp_actD) 1); by (asm_simp_tac (simpset() addsimps [split_box_conj,box_stp_act]) 1); by (etac (temp_use INV1) 1); by (rtac (temp_use prem3) 1); by (asm_full_simp_tac (simpset() addsimps [split_box_conj,temp_use NotDmd,not_angle]) 1); qed "WF_leadsto"; val prems = goal (the_context ()) "[| |- $P & N --> P` | Q`; \ \ |- ($P & N) & <A>_v --> Q`; \ \ |- []P & []N & []F --> <>Enabled(<A>_v) |] \ \ ==> |- []N & SF(A)_v & []F --> (P ~> Q)"; by (clarsimp_tac (temp_css addSDs2 [BoxSFI]) 1); by (rtac (temp_use ensures) 1); by (TRYALL (ares_tac prems)); by (eres_inst_tac [("F","F")] dup_boxE 1); by (merge_temp_box_tac 1); by (etac STL4Edup 1); by (atac 1); by (clarsimp_tac (temp_css addsimps2 [SF_def]) 1); by (rtac (temp_use STL2) 1); by (etac mp 1); by (resolve_tac (map temp_use (prems RL [STL4])) 1); by (asm_simp_tac (simpset() addsimps [split_box_conj, STL3]) 1); qed "SF1"; val [prem1,prem2,prem3,prem4] = goal (the_context ()) "[| |- N & <B>_f --> <M>_g; \ \ |- $P & P` & <N & A>_f --> B; \ \ |- P & Enabled(<M>_g) --> Enabled(<A>_f); \ \ |- [](N & [~B]_f) & WF(A)_f & []F & <>[]Enabled(<M>_g) --> <>[]P |] \ \ ==> |- []N & WF(A)_f & []F --> WF(M)_g"; by (clarsimp_tac (temp_css addSDs2 [BoxWFI, (temp_use BoxDmdBox) RS iffD2] addsimps2 [read_instantiate [("A","M")] WF_def]) 1); by (eres_inst_tac [("F","F")] dup_boxE 1); by (merge_temp_box_tac 1); by (etac STL4Edup 1); by (atac 1); by (clarsimp_tac (temp_css addSIs2 [(temp_use BoxDmd_simple) RS (temp_use (prem1 RS DmdImpl))]) 1); by (rtac classical 1); by (subgoal_tac "sigmaa |= <>(($P & P` & N) & <A>_f)" 1); by (force_tac (temp_css addsimps2 [angle_def] addSIs2 [prem2] addSEs2 [DmdImplE]) 1); by (rtac (temp_use (rewrite_rule [temp_rewrite DmdDmd] (BoxDmd_simple RS DmdImpl))) 1); by (asm_full_simp_tac (simpset() addsimps [temp_use NotDmd, not_angle]) 1); by (merge_act_box_tac 1); by (forward_tac [temp_use prem4] 1); by (TRYALL atac); by (dtac (temp_use STL6) 1); by (atac 1); by (eres_inst_tac [("V","sigmaa |= <>[]P")] thin_rl 1); by (eres_inst_tac [("V","sigmaa |= []F")] thin_rl 1); by (dtac (temp_use BoxWFI) 1); by (eres_inst_tac [("F", "ACT N & [~B]_f")] dup_boxE 1); by (merge_temp_box_tac 1); by (etac DmdImpldup 1); by (atac 1); by (auto_tac (temp_css addsimps2 [split_box_conj,STL3,WF_Box,box_stp_act])); by (force_tac (temp_css addSEs2 [read_instantiate [("P","P")] TLA2E]) 1); by (rtac (temp_use STL2) 1); by (force_tac (temp_css addsimps2 [WF_def,split_box_conj] addSEs2 [mp] addSIs2 [InitDmd, prem3 RS STL4]) 1); qed "WF2"; val [prem1,prem2,prem3,prem4] = goal (the_context ()) "[| |- N & <B>_f --> <M>_g; \ \ |- $P & P` & <N & A>_f --> B; \ \ |- P & Enabled(<M>_g) --> Enabled(<A>_f); \ \ |- [](N & [~B]_f) & SF(A)_f & []F & []<>Enabled(<M>_g) --> <>[]P |] \ \ ==> |- []N & SF(A)_f & []F --> SF(M)_g"; by (clarsimp_tac (temp_css addSDs2 [BoxSFI] addsimps2 [read_instantiate [("A","M")] SF_def]) 1); by (eres_inst_tac [("F","F")] dup_boxE 1); by (eres_inst_tac [("F","TEMP <>Enabled(<M>_g)")] dup_boxE 1); by (merge_temp_box_tac 1); by (etac STL4Edup 1); by (atac 1); by (clarsimp_tac (temp_css addSIs2 [(temp_use BoxDmd_simple) RS (temp_use (prem1 RS DmdImpl))]) 1); by (rtac classical 1); by (subgoal_tac "sigmaa |= <>(($P & P` & N) & <A>_f)" 1); by (force_tac (temp_css addsimps2 [angle_def] addSIs2 [prem2] addSEs2 [DmdImplE]) 1); by (rtac (temp_use (rewrite_rule [temp_rewrite DmdDmd] (BoxDmd_simple RS DmdImpl))) 1); by (asm_full_simp_tac (simpset() addsimps [temp_use NotDmd, not_angle]) 1); by (merge_act_box_tac 1); by (forward_tac [temp_use prem4] 1); by (TRYALL atac); by (eres_inst_tac [("V","sigmaa |= []F")] thin_rl 1); by (dtac (temp_use BoxSFI) 1); by (eres_inst_tac [("F","TEMP <>Enabled(<M>_g)")] dup_boxE 1); by (eres_inst_tac [("F", "ACT N & [~B]_f")] dup_boxE 1); by (merge_temp_box_tac 1); by (etac DmdImpldup 1); by (atac 1); by (auto_tac (temp_css addsimps2 [split_box_conj,STL3,SF_Box,box_stp_act])); by (force_tac (temp_css addSEs2 [read_instantiate [("P","P")] TLA2E]) 1); by (rtac (temp_use STL2) 1); by (force_tac (temp_css addsimps2 [SF_def,split_box_conj] addSEs2 [mp,InfImpl] addSIs2 [prem3]) 1); qed "SF2"; (* ------------------------------------------------------------------------- *) (*** Liveness proofs by well-founded orderings ***) (* ------------------------------------------------------------------------- *) section "Well-founded orderings"; val p1::prems = goal (the_context ()) "[| wf r; \ \ !!x. sigma |= F x ~> (G | (EX y. #((y,x):r) & F y)) \ \ |] ==> sigma |= F x ~> G"; by (rtac (p1 RS wf_induct) 1); by (rtac (temp_use LatticeTriangle) 1); by (resolve_tac prems 1); by (auto_tac (temp_css addsimps2 [leadsto_exists])); by (case_tac "(y,x):r" 1); by (Force_tac 1); by (force_tac (temp_css addsimps2 leadsto_def::Init_simps addSIs2 [necT]) 1); qed "wf_leadsto"; (* If r is well-founded, state function v cannot decrease forever *) Goal "!!r. wf r ==> |- [][ (v`, $v) : #r ]_v --> <>[][#False]_v"; by (Clarsimp_tac 1); by (rtac ccontr 1); by (subgoal_tac "sigma |= (EX x. v=#x) ~> #False" 1); by (dtac ((temp_use leadsto_false) RS iffD1 RS (temp_use STL2_gen)) 1); by (force_tac (temp_css addsimps2 Init_defs) 1); by (clarsimp_tac (temp_css addsimps2 [leadsto_exists,not_square]@more_temp_simps) 1); by (etac wf_leadsto 1); by (rtac (temp_use ensures_simple) 1); by (TRYALL atac); by (auto_tac (temp_css addsimps2 [square_def,angle_def])); qed "wf_not_box_decrease"; (* "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. *) val [prem] = goal (the_context ()) "wf r ==> |- []<>((v`, $v) : #r) --> []<><(v`, $v) ~: #r>_v"; by (Clarsimp_tac 1); by (rtac ccontr 1); by (asm_full_simp_tac (simpset() addsimps not_angle::more_temp_simps) 1); by (dtac (prem RS (temp_use wf_not_dmd_box_decrease)) 1); by (dtac (temp_use BoxDmdDmdBox) 1); by (atac 1); by (subgoal_tac "sigma |= []<>((#False)::action)" 1); by (Force_tac 1); by (etac STL4E 1); by (rtac DmdImpl 1); by (force_tac (temp_css addIs2 [prem RS wf_irrefl]) 1); qed "wf_box_dmd_decrease"; (* In particular, for natural numbers, if n decreases infinitely often then it has to increase infinitely often. *) Goal "!!n::nat stfun. |- []<>(n` < $n) --> []<>($n < n`)"; by (Clarsimp_tac 1); by (subgoal_tac "sigma |= []<><~( (n`,$n) : #less_than )>_n" 1); by (etac thin_rl 1); by (etac STL4E 1); by (rtac DmdImpl 1); by (clarsimp_tac (temp_css addsimps2 [angle_def]) 1); by (rtac (temp_use wf_box_dmd_decrease) 1); by (auto_tac (temp_css addSEs2 [STL4E,DmdImplE])); qed "nat_box_dmd_decrease"; (* ------------------------------------------------------------------------- *) (*** Flexible quantification over state variables ***) (* ------------------------------------------------------------------------- *) section "Flexible quantification"; val [prem1,prem2] = goal (the_context ()) "[| basevars vs; (!!x. basevars (x,vs) ==> sigma |= F x) |]\ \ ==> sigma |= (AALL x. F x)"; by (auto_tac (temp_css addsimps2 [aall_def] addSEs2 [eexE] addSIs2 [prem1] addSDs2 [prem2])); qed "aallI"; Goalw [aall_def] "|- (AALL x. F x) --> F x"; by (Clarsimp_tac 1); by (etac contrapos_np 1); by (force_tac (temp_css addSIs2 [eexI]) 1); qed "aallE"; (* monotonicity of quantification *) val [min,maj] = goal (the_context ()) "[| sigma |= EEX x. F x; !!x. sigma |= F x --> G x |] ==> sigma |= EEX x. G x"; by (rtac (unit_base RS (min RS eexE)) 1); by (rtac (temp_use eexI) 1); by (etac ((rewrite_rule intensional_rews maj) RS mp) 1); qed "eex_mono"; val [min,maj] = goal (the_context ()) "[| sigma |= AALL x. F(x); !!x. sigma |= F(x) --> G(x) |] ==> sigma |= AALL x. G(x)"; by (rtac (unit_base RS aallI) 1); by (rtac ((rewrite_rule intensional_rews maj) RS mp) 1); by (rtac (min RS (temp_use aallE)) 1); qed "aall_mono"; (* Derived history introduction rule *) val [p1,p2,p3,p4,p5] = goal (the_context ()) "[| 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)"; by (rtac ((temp_use history) RS eexE) 1); by (rtac p3 1); by (rtac (temp_use eexI) 1); by (Clarsimp_tac 1); by (rtac conjI 1); by (cut_facts_tac [p2] 2); by (merge_box_tac 2); by (force_tac (temp_css addSEs2 [STL4E,p5]) 2); by (cut_facts_tac [p1] 1); by (force_tac (temp_css addsimps2 Init_defs addSEs2 [p4]) 1); qed "historyI"; (* ---------------------------------------------------------------------- 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 **) ---------------------------------------------------------------------- *)