(* Title: HOL/TLA/TLA.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
*)
section \<open>The temporal level of TLA\<close>
theory TLA
imports Init
begin
consts
(** abstract syntax **)
Box :: "('w::world) form \<Rightarrow> temporal"
Dmd :: "('w::world) form \<Rightarrow> temporal"
leadsto :: "['w::world form, 'v::world form] \<Rightarrow> temporal"
Stable :: "stpred \<Rightarrow> temporal"
WF :: "[action, 'a stfun] \<Rightarrow> temporal"
SF :: "[action, 'a stfun] \<Rightarrow> temporal"
(* Quantification over (flexible) state variables *)
EEx :: "('a stfun \<Rightarrow> temporal) \<Rightarrow> temporal" (binder "Eex " 10)
AAll :: "('a stfun \<Rightarrow> temporal) \<Rightarrow> temporal" (binder "Aall " 10)
(** concrete syntax **)
syntax
"_Box" :: "lift \<Rightarrow> lift" ("(\<box>_)" [40] 40)
"_Dmd" :: "lift \<Rightarrow> lift" ("(\<diamond>_)" [40] 40)
"_leadsto" :: "[lift,lift] \<Rightarrow> lift" ("(_ \<leadsto> _)" [23,22] 22)
"_stable" :: "lift \<Rightarrow> lift" ("(stable/ _)")
"_WF" :: "[lift,lift] \<Rightarrow> lift" ("(WF'(_')'_(_))" [0,60] 55)
"_SF" :: "[lift,lift] \<Rightarrow> lift" ("(SF'(_')'_(_))" [0,60] 55)
"_EEx" :: "[idts, lift] \<Rightarrow> lift" ("(3\<exists>\<exists> _./ _)" [0,10] 10)
"_AAll" :: "[idts, lift] \<Rightarrow> lift" ("(3\<forall>\<forall> _./ _)" [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 \<Turnstile> \<box>F" <= "_Box F sigma"
"sigma \<Turnstile> \<diamond>F" <= "_Dmd F sigma"
"sigma \<Turnstile> F \<leadsto> G" <= "_leadsto F G sigma"
"sigma \<Turnstile> stable P" <= "_stable P sigma"
"sigma \<Turnstile> WF(A)_v" <= "_WF A v sigma"
"sigma \<Turnstile> SF(A)_v" <= "_SF A v sigma"
"sigma \<Turnstile> \<exists>\<exists>x. F" <= "_EEx x F sigma"
"sigma \<Turnstile> \<forall>\<forall>x. F" <= "_AAll x F sigma"
axiomatization where
(* Definitions of derived operators *)
dmd_def: "\<And>F. TEMP \<diamond>F == TEMP \<not>\<box>\<not>F"
axiomatization where
boxInit: "\<And>F. TEMP \<box>F == TEMP \<box>Init F" and
leadsto_def: "\<And>F G. TEMP F \<leadsto> G == TEMP \<box>(Init F \<longrightarrow> \<diamond>G)" and
stable_def: "\<And>P. TEMP stable P == TEMP \<box>($P \<longrightarrow> P$)" and
WF_def: "TEMP WF(A)_v == TEMP \<diamond>\<box> Enabled(<A>_v) \<longrightarrow> \<box>\<diamond><A>_v" and
SF_def: "TEMP SF(A)_v == TEMP \<box>\<diamond> Enabled(<A>_v) \<longrightarrow> \<box>\<diamond><A>_v" and
aall_def: "TEMP (\<forall>\<forall>x. F x) == TEMP \<not> (\<exists>\<exists>x. \<not> F x)"
axiomatization where
(* Base axioms for raw TLA. *)
normalT: "\<And>F G. \<turnstile> \<box>(F \<longrightarrow> G) \<longrightarrow> (\<box>F \<longrightarrow> \<box>G)" and (* polymorphic *)
reflT: "\<And>F. \<turnstile> \<box>F \<longrightarrow> F" and (* F::temporal *)
transT: "\<And>F. \<turnstile> \<box>F \<longrightarrow> \<box>\<box>F" and (* polymorphic *)
linT: "\<And>F G. \<turnstile> \<diamond>F \<and> \<diamond>G \<longrightarrow> (\<diamond>(F \<and> \<diamond>G)) \<or> (\<diamond>(G \<and> \<diamond>F))" and
discT: "\<And>F. \<turnstile> \<box>(F \<longrightarrow> \<diamond>(\<not>F \<and> \<diamond>F)) \<longrightarrow> (F \<longrightarrow> \<box>\<diamond>F)" and
primeI: "\<And>P. \<turnstile> \<box>P \<longrightarrow> Init P`" and
primeE: "\<And>P F. \<turnstile> \<box>(Init P \<longrightarrow> \<box>F) \<longrightarrow> Init P` \<longrightarrow> (F \<longrightarrow> \<box>F)" and
indT: "\<And>P F. \<turnstile> \<box>(Init P \<and> \<not>\<box>F \<longrightarrow> Init P` \<and> F) \<longrightarrow> Init P \<longrightarrow> \<box>F" and
allT: "\<And>F. \<turnstile> (\<forall>x. \<box>(F x)) = (\<box>(\<forall> x. F x))"
axiomatization where
necT: "\<And>F. \<turnstile> F \<Longrightarrow> \<turnstile> \<box>F" (* polymorphic *)
axiomatization where
(* Flexible quantification: refinement mappings, history variables *)
eexI: "\<turnstile> F x \<longrightarrow> (\<exists>\<exists>x. F x)" and
eexE: "\<lbrakk> sigma \<Turnstile> (\<exists>\<exists>x. F x); basevars vs;
(\<And>x. \<lbrakk> basevars (x, vs); sigma \<Turnstile> F x \<rbrakk> \<Longrightarrow> (G sigma)::bool)
\<rbrakk> \<Longrightarrow> G sigma" and
history: "\<turnstile> \<exists>\<exists>h. Init(h = ha) \<and> \<box>(\<forall>x. $h = #x \<longrightarrow> h` = hb x)"
(* Specialize intensional introduction/elimination rules for temporal formulas *)
lemma tempI [intro!]: "(\<And>sigma. sigma \<Turnstile> (F::temporal)) \<Longrightarrow> \<turnstile> F"
apply (rule intI)
apply (erule meta_spec)
done
lemma tempD [dest]: "\<turnstile> (F::temporal) \<Longrightarrow> sigma \<Turnstile> F"
by (erule intD)
(* ======== Functions to "unlift" temporal theorems ====== *)
ML \<open>
(* 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 ctxt th =
(rewrite_rule ctxt @{thms action_rews} (th RS @{thm tempD}))
handle THM _ => action_unlift ctxt th;
(* Turn \<turnstile> F = G into meta-level rewrite rule F == G *)
val temp_rewrite = int_rewrite
fun temp_use ctxt th =
case Thm.concl_of th of
Const _ $ (Const (@{const_name Intensional.Valid}, _) $ _) =>
((flatten (temp_unlift ctxt th)) handle THM _ => th)
| _ => th;
fun try_rewrite ctxt th = temp_rewrite ctxt th handle THM _ => temp_use ctxt th;
\<close>
attribute_setup temp_unlift =
\<open>Scan.succeed (Thm.rule_attribute (temp_unlift o Context.proof_of))\<close>
attribute_setup temp_rewrite =
\<open>Scan.succeed (Thm.rule_attribute (temp_rewrite o Context.proof_of))\<close>
attribute_setup temp_use =
\<open>Scan.succeed (Thm.rule_attribute (temp_use o Context.proof_of))\<close>
attribute_setup try_rewrite =
\<open>Scan.succeed (Thm.rule_attribute (try_rewrite o Context.proof_of))\<close>
(* ------------------------------------------------------------------------- *)
(*** "Simple temporal logic": only \<box> and \<diamond> ***)
(* ------------------------------------------------------------------------- *)
section "Simple temporal logic"
(* \<box>\<not>F == \<box>\<not>Init F *)
lemmas boxNotInit = boxInit [of "LIFT \<not>F", unfolded Init_simps] for F
lemma dmdInit: "TEMP \<diamond>F == TEMP \<diamond> Init F"
apply (unfold dmd_def)
apply (unfold boxInit [of "LIFT \<not>F"])
apply (simp (no_asm) add: Init_simps)
done
lemmas dmdNotInit = dmdInit [of "LIFT \<not>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: "\<turnstile> \<box>F \<longrightarrow> Init F"
apply (unfold boxInit [of F])
apply (rule STL2)
done
(* see also STL2_pr below: "\<turnstile> \<box>P \<longrightarrow> Init P & Init (P`)" *)
(* Dual versions for \<diamond> *)
lemma InitDmd: "\<turnstile> F \<longrightarrow> \<diamond> F"
apply (unfold dmd_def)
apply (auto dest!: STL2 [temp_use])
done
lemma InitDmd_gen: "\<turnstile> Init F \<longrightarrow> \<diamond>F"
apply clarsimp
apply (drule InitDmd [temp_use])
apply (simp add: dmdInitD)
done
(* ------------------------ STL3 ------------------------------------------- *)
lemma STL3: "\<turnstile> (\<box>\<box>F) = (\<box>F)"
by (auto elim: transT [temp_use] STL2 [temp_use])
(* corresponding elimination rule introduces double boxes:
\<lbrakk> (sigma \<Turnstile> \<box>F); (sigma \<Turnstile> \<box>\<box>F) \<Longrightarrow> PROP W \<rbrakk> \<Longrightarrow> PROP W
*)
lemmas dup_boxE = STL3 [temp_unlift, THEN iffD2, elim_format]
lemmas dup_boxD = STL3 [temp_unlift, THEN iffD1]
(* dual versions for \<diamond> *)
lemma DmdDmd: "\<turnstile> (\<diamond>\<diamond>F) = (\<diamond>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 "\<turnstile> F \<longrightarrow> G"
shows "\<turnstile> \<box>F \<longrightarrow> \<box>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: "\<lbrakk> sigma \<Turnstile> \<box>F; \<turnstile> F \<longrightarrow> G \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>G"
by (erule (1) STL4 [temp_use])
lemma STL4_gen: "\<turnstile> Init F \<longrightarrow> Init G \<Longrightarrow> \<turnstile> \<box>F \<longrightarrow> \<box>G"
apply (drule STL4)
apply (simp add: boxInitD)
done
lemma STL4E_gen: "\<lbrakk> sigma \<Turnstile> \<box>F; \<turnstile> Init F \<longrightarrow> Init G \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>G"
by (erule (1) STL4_gen [temp_use])
(* see also STL4Edup below, which allows an auxiliary boxed formula:
\<box>A /\ F => G
-----------------
\<box>A /\ \<box>F => \<box>G
*)
(* The dual versions for \<diamond> *)
lemma DmdImpl:
assumes prem: "\<turnstile> F \<longrightarrow> G"
shows "\<turnstile> \<diamond>F \<longrightarrow> \<diamond>G"
apply (unfold dmd_def)
apply (fastforce intro!: prem [temp_use] elim!: STL4E [temp_use])
done
lemma DmdImplE: "\<lbrakk> sigma \<Turnstile> \<diamond>F; \<turnstile> F \<longrightarrow> G \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<diamond>G"
by (erule (1) DmdImpl [temp_use])
(* ------------------------ STL5 ------------------------------------------- *)
lemma STL5: "\<turnstile> (\<box>F \<and> \<box>G) = (\<box>(F \<and> G))"
apply auto
apply (subgoal_tac "sigma \<Turnstile> \<box> (G \<longrightarrow> (F \<and> 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 \<Turnstile> \<box>F"
and "sigma \<Turnstile> \<box>G"
and "sigma \<Turnstile> \<box>(F\<and>G) \<Longrightarrow> 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: "\<lbrakk> sigma \<Turnstile> \<box>F; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
ML \<open>
fun merge_box_tac ctxt i =
REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE} i, assume_tac ctxt i,
eresolve_tac ctxt @{thms box_thin} i])
fun merge_temp_box_tac ctxt i =
REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE_temp} i, assume_tac ctxt i,
Rule_Insts.eres_inst_tac ctxt [((("'a", 0), Position.none), "behavior")] [] @{thm box_thin} i])
fun merge_stp_box_tac ctxt i =
REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE_stp} i, assume_tac ctxt i,
Rule_Insts.eres_inst_tac ctxt [((("'a", 0), Position.none), "state")] [] @{thm box_thin} i])
fun merge_act_box_tac ctxt i =
REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE_act} i, assume_tac ctxt i,
Rule_Insts.eres_inst_tac ctxt [((("'a", 0), Position.none), "state * state")] [] @{thm box_thin} i])
\<close>
method_setup merge_box = \<open>Scan.succeed (SIMPLE_METHOD' o merge_box_tac)\<close>
method_setup merge_temp_box = \<open>Scan.succeed (SIMPLE_METHOD' o merge_temp_box_tac)\<close>
method_setup merge_stp_box = \<open>Scan.succeed (SIMPLE_METHOD' o merge_stp_box_tac)\<close>
method_setup merge_act_box = \<open>Scan.succeed (SIMPLE_METHOD' o merge_act_box_tac)\<close>
(* rewrite rule to push universal quantification through box:
(sigma \<Turnstile> \<box>(\<forall>x. F x)) = (\<forall>x. (sigma \<Turnstile> \<box>F x))
*)
lemmas all_box = allT [temp_unlift, symmetric]
lemma DmdOr: "\<turnstile> (\<diamond>(F \<or> G)) = (\<diamond>F \<or> \<diamond>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: "\<turnstile> (\<exists>x. \<diamond>(F x)) = (\<diamond>(\<exists>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: "\<And>sigma. \<lbrakk> sigma \<Turnstile> \<box>A; sigma \<Turnstile> \<box>F; \<turnstile> F \<and> \<box>A \<longrightarrow> G \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>G"
apply (erule dup_boxE)
apply merge_box
apply (erule STL4E)
apply assumption
done
lemma DmdImpl2:
"\<And>sigma. \<lbrakk> sigma \<Turnstile> \<diamond>F; sigma \<Turnstile> \<box>(F \<longrightarrow> G) \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<diamond>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 \<Turnstile> \<box>\<diamond>F"
and 2: "sigma \<Turnstile> \<box>G"
and 3: "\<turnstile> F \<and> G \<longrightarrow> H"
shows "sigma \<Turnstile> \<box>\<diamond>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: "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(\<box>F \<and> 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: "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(F \<and> G)"
apply (unfold dmd_def)
apply clarsimp
apply merge_box
apply (fastforce elim!: notE STL4E [temp_use])
done
lemma BoxDmd2_simple: "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(G \<and> F)"
apply (unfold dmd_def)
apply clarsimp
apply merge_box
apply (fastforce elim!: notE STL4E [temp_use])
done
lemma DmdImpldup:
assumes 1: "sigma \<Turnstile> \<box>A"
and 2: "sigma \<Turnstile> \<diamond>F"
and 3: "\<turnstile> \<box>A \<and> F \<longrightarrow> G"
shows "sigma \<Turnstile> \<diamond>G"
apply (rule 2 [THEN 1 [THEN BoxDmd [temp_use]], THEN DmdImplE])
apply (rule 3)
done
lemma STL6: "\<turnstile> \<diamond>\<box>F \<and> \<diamond>\<box>G \<longrightarrow> \<diamond>\<box>(F \<and> 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: "\<turnstile> (\<box>#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: "\<turnstile> (\<diamond>#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: "\<turnstile> (\<not>\<box>F) = (\<diamond>\<not>F)"
by (simp add: dmd_def)
lemma NotDmd: "\<turnstile> (\<not>\<diamond>F) = (\<box>\<not>F)"
by (simp add: dmd_def)
(* These are not declared by default, because they could be harmful,
e.g. \<box>F & \<not>\<box>F becomes \<box>F & \<diamond>\<not>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: "\<turnstile> (\<box>\<diamond>\<box>F) = (\<diamond>\<box>F)"
apply (auto dest!: STL2 [temp_use])
apply (rule ccontr)
apply (subgoal_tac "sigma \<Turnstile> \<diamond>\<box>\<box>F \<and> \<diamond>\<box>\<not>\<box>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: "\<turnstile> (\<diamond>\<box>\<diamond>F) = (\<box>\<diamond>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: "\<And>sigma. \<lbrakk> sigma \<Turnstile> \<box>F \<or> \<box>G \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>(F \<or> G)"
by (fastforce elim!: STL4E [temp_use])
(* "persistently implies infinitely often" *)
lemma DBImplBD: "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<box>\<diamond>F"
apply clarsimp
apply (rule ccontr)
apply (simp add: more_temp_simps2)
apply (drule STL6 [temp_use])
apply assumption
apply simp
done
lemma BoxDmdDmdBox: "\<turnstile> \<box>\<diamond>F \<and> \<diamond>\<box>G \<longrightarrow> \<box>\<diamond>(F \<and> G)"
apply clarsimp
apply (rule ccontr)
apply (unfold more_temp_simps2)
apply (drule STL6 [temp_use])
apply assumption
apply (subgoal_tac "sigma \<Turnstile> \<diamond>\<box>\<not>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: "\<turnstile> \<box>P \<longrightarrow> Init P \<and> Init P`"
by (fastforce intro!: STL2_gen [temp_use] primeI [temp_use])
(* Auxiliary lemma allows priming of boxed actions *)
lemma BoxPrime: "\<turnstile> \<box>P \<longrightarrow> \<box>($P \<and> P$)"
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 "\<turnstile> $P \<and> P$ \<longrightarrow> A"
shows "\<turnstile> \<box>P \<longrightarrow> \<box>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: "\<lbrakk> sigma \<Turnstile> \<box>P; \<turnstile> $P \<and> P$ \<longrightarrow> A \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>A"
by (erule (1) TLA2 [temp_use])
lemma DmdPrime: "\<turnstile> (\<diamond>P`) \<longrightarrow> (\<diamond>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:
"\<lbrakk> sigma \<Turnstile> \<box>H; sigma \<Turnstile> Init P; \<turnstile> H \<longrightarrow> (Init P \<and> \<not>\<box>F \<longrightarrow> Init(P`) \<and> F) \<rbrakk>
\<Longrightarrow> sigma \<Turnstile> \<box>F"
apply (rule indT [temp_use])
apply (erule (2) STL4E)
done
lemma box_stp_act: "\<turnstile> (\<box>$P) = (\<box>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:
"\<turnstile> (Init P) \<longrightarrow> (stable P) \<longrightarrow> \<box>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:
"\<And>P. \<turnstile> $P \<and> A \<longrightarrow> P` \<Longrightarrow> \<turnstile> \<box>A \<longrightarrow> stable P"
apply (unfold stable_def)
apply (fastforce elim!: STL4E [temp_use])
done
lemma Stable: "\<lbrakk> sigma \<Turnstile> \<box>A; \<turnstile> $P \<and> A \<longrightarrow> P` \<rbrakk> \<Longrightarrow> sigma \<Turnstile> stable P"
by (erule (1) StableT [temp_use])
(* Generalization of INV1 *)
lemma StableBox: "\<turnstile> (stable P) \<longrightarrow> \<box>(Init P \<longrightarrow> \<box>P)"
apply (unfold stable_def)
apply clarsimp
apply (erule dup_boxE)
apply (force simp: stable_def elim: STL4E [temp_use] INV1 [temp_use])
done
lemma DmdStable: "\<turnstile> (stable P) \<and> \<diamond>P \<longrightarrow> \<diamond>\<box>P"
apply clarsimp
apply (rule DmdImpl2)
prefer 2
apply (erule StableBox [temp_use])
apply (simp add: dmdInitD)
done
(* ---------------- (Semi-)automatic invariant tactics ---------------------- *)
ML \<open>
(* inv_tac reduces goals of the form ... \<Longrightarrow> sigma \<Turnstile> \<box>P *)
fun inv_tac ctxt =
SELECT_GOAL
(EVERY
[auto_tac ctxt,
TRY (merge_box_tac ctxt 1),
resolve_tac ctxt [temp_use ctxt @{thm INV1}] 1, (* fail if the goal is not a box *)
TRYALL (eresolve_tac ctxt @{thms Stable})]);
(* auto_inv_tac applies inv_tac and then tries to attack the subgoals
in simple cases it may be able to handle goals like \<turnstile> MyProg \<longrightarrow> \<box>Inv.
In these simple cases the simplifier seems to be more useful than the
auto-tactic, which applies too much propositional logic and simplifies
too late.
*)
fun auto_inv_tac ctxt =
SELECT_GOAL
(inv_tac ctxt 1 THEN
(TRYALL (action_simp_tac
(ctxt addsimps [@{thm Init_stp}, @{thm Init_act}]) [] [@{thm squareE}])));
\<close>
method_setup invariant = \<open>
Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o inv_tac))
\<close>
method_setup auto_invariant = \<open>
Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o auto_inv_tac))
\<close>
lemma unless: "\<turnstile> \<box>($P \<longrightarrow> P` \<or> Q`) \<longrightarrow> (stable P) \<or> \<diamond>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 \<box> and \<diamond> for state predicates *)
lemma BoxRec: "\<turnstile> (\<box>P) = (Init P \<and> \<box>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: "\<turnstile> (\<diamond>P) = (Init P \<or> \<diamond>P`)"
apply (unfold dmd_def BoxRec [temp_rewrite])
apply (auto simp: Init_simps)
done
lemma DmdRec2: "\<And>sigma. \<lbrakk> sigma \<Turnstile> \<diamond>P; sigma \<Turnstile> \<box>\<not>P` \<rbrakk> \<Longrightarrow> sigma \<Turnstile> Init P"
apply (force simp: DmdRec [temp_rewrite] dmd_def)
done
lemma InfinitePrime: "\<turnstile> (\<box>\<diamond>P) = (\<box>\<diamond>P`)"
apply auto
apply (rule classical)
apply (rule DBImplBD [temp_use])
apply (subgoal_tac "sigma \<Turnstile> \<diamond>\<box>P")
apply (fastforce elim!: DmdImplE [temp_use] TLA2E [temp_use])
apply (subgoal_tac "sigma \<Turnstile> \<diamond>\<box> (\<diamond>P \<and> \<box>\<not>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:
"\<lbrakk> sigma \<Turnstile> \<box>N; sigma \<Turnstile> \<box>\<diamond>A; \<turnstile> A \<and> N \<longrightarrow> P` \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>\<diamond>P"
apply (unfold InfinitePrime [temp_rewrite])
apply (rule InfImpl)
apply assumption+
done
(* ------------------------ fairness ------------------------------------------- *)
section "fairness"
(* alternative definitions of fairness *)
lemma WF_alt: "\<turnstile> WF(A)_v = (\<box>\<diamond>\<not>Enabled(<A>_v) \<or> \<box>\<diamond><A>_v)"
apply (unfold WF_def dmd_def)
apply fastforce
done
lemma SF_alt: "\<turnstile> SF(A)_v = (\<diamond>\<box>\<not>Enabled(<A>_v) \<or> \<box>\<diamond><A>_v)"
apply (unfold SF_def dmd_def)
apply fastforce
done
(* theorems to "box" fairness conditions *)
lemma BoxWFI: "\<turnstile> WF(A)_v \<longrightarrow> \<box>WF(A)_v"
by (auto simp: WF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
lemma WF_Box: "\<turnstile> (\<box>WF(A)_v) = WF(A)_v"
by (fastforce intro!: BoxWFI [temp_use] dest!: STL2 [temp_use])
lemma BoxSFI: "\<turnstile> SF(A)_v \<longrightarrow> \<box>SF(A)_v"
by (auto simp: SF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
lemma SF_Box: "\<turnstile> (\<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: "\<turnstile> SF(A)_v \<longrightarrow> 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 \<open>
fun box_fair_tac ctxt =
SELECT_GOAL (REPEAT (dresolve_tac ctxt [@{thm BoxWFI}, @{thm BoxSFI}] 1))
\<close>
(* ------------------------------ leads-to ------------------------------ *)
section "\<leadsto>"
lemma leadsto_init: "\<turnstile> (Init F) \<and> (F \<leadsto> G) \<longrightarrow> \<diamond>G"
apply (unfold leadsto_def)
apply (auto dest!: STL2 [temp_use])
done
(* \<turnstile> F & (F \<leadsto> G) \<longrightarrow> \<diamond>G *)
lemmas leadsto_init_temp = leadsto_init [where 'a = behavior, unfolded Init_simps]
lemma streett_leadsto: "\<turnstile> (\<box>\<diamond>Init F \<longrightarrow> \<box>\<diamond>G) = (\<diamond>(F \<leadsto> 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 \<Turnstile> \<box>\<diamond>\<diamond>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: "\<turnstile> \<box>\<diamond>F \<and> (F \<leadsto> G) \<longrightarrow> \<box>\<diamond>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: "\<turnstile> (Enabled(<A>_v) \<leadsto> <A>_v) \<longrightarrow> SF(A)_v"
apply (unfold SF_def)
apply (clarsimp elim!: leadsto_infinite [temp_use])
done
lemma leadsto_WF: "\<turnstile> (Enabled(<A>_v) \<leadsto> <A>_v) \<longrightarrow> WF(A)_v"
by (clarsimp intro!: SFImplWF [temp_use] leadsto_SF [temp_use])
(* introduce an invariant into the proof of a leadsto assertion.
\<box>I \<longrightarrow> ((P \<leadsto> Q) = (P /\ I \<leadsto> Q))
*)
lemma INV_leadsto: "\<turnstile> \<box>I \<and> (P \<and> I \<leadsto> Q) \<longrightarrow> (P \<leadsto> 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: "\<turnstile> (Init F \<and> \<box>\<not>G \<leadsto> G) \<longrightarrow> (F \<leadsto> G)"
apply (unfold leadsto_def dmd_def)
apply (force simp: Init_simps elim!: STL4E [temp_use])
done
lemma leadsto_false: "\<turnstile> (F \<leadsto> #False) = (\<box>\<not>F)"
apply (unfold leadsto_def)
apply (simp add: boxNotInitD)
done
lemma leadsto_exists: "\<turnstile> ((\<exists>x. F x) \<leadsto> G) = (\<forall>x. (F x \<leadsto> 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: "\<turnstile> \<box>(Init F \<longrightarrow> Init G) \<longrightarrow> (F \<leadsto> 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: "\<And>F G. \<turnstile> F \<longrightarrow> G \<Longrightarrow> \<turnstile> F \<leadsto> G"
by (auto simp: Init_def intro!: ImplLeadsto_gen [temp_use] necT [temp_use])
lemma EnsuresLeadsto:
assumes "\<turnstile> A \<and> $P \<longrightarrow> Q`"
shows "\<turnstile> \<box>A \<longrightarrow> (P \<leadsto> Q)"
apply (unfold leadsto_def)
apply (clarsimp elim!: INV_leadsto [temp_use])
apply (erule STL4E_gen)
apply (auto simp: Init_defs intro!: PrimeDmd [temp_use] assms [temp_use])
done
lemma EnsuresLeadsto2: "\<turnstile> \<box>($P \<longrightarrow> Q`) \<longrightarrow> (P \<leadsto> 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: "\<turnstile> $P \<and> N \<longrightarrow> P` \<or> Q`"
and 2: "\<turnstile> ($P \<and> N) \<and> A \<longrightarrow> Q`"
shows "\<turnstile> \<box>N \<and> \<box>(\<box>P \<longrightarrow> \<diamond>A) \<longrightarrow> (P \<leadsto> Q)"
apply (unfold leadsto_def)
apply clarsimp
apply (erule STL4Edup)
apply assumption
apply clarsimp
apply (subgoal_tac "sigmaa \<Turnstile> \<box>($P \<longrightarrow> P` \<or> 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:
"\<lbrakk> \<turnstile> $P \<and> N \<longrightarrow> P` \<or> Q`;
\<turnstile> ($P \<and> N) \<and> A \<longrightarrow> Q`
\<rbrakk> \<Longrightarrow> \<turnstile> \<box>N \<and> \<box>\<diamond>A \<longrightarrow> (P \<leadsto> Q)"
apply clarsimp
apply (erule (2) ensures [temp_use])
apply (force elim!: STL4E [temp_use])
done
lemma EnsuresInfinite:
"\<lbrakk> sigma \<Turnstile> \<box>\<diamond>P; sigma \<Turnstile> \<box>A; \<turnstile> A \<and> $P \<longrightarrow> Q` \<rbrakk> \<Longrightarrow> sigma \<Turnstile> \<box>\<diamond>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: "\<turnstile> F \<leadsto> F"
apply (unfold leadsto_def)
apply (rule necT InitDmd_gen)+
done
lemma LatticeTransitivity: "\<turnstile> (G \<leadsto> H) \<and> (F \<leadsto> G) \<longrightarrow> (F \<leadsto> H)"
apply (unfold leadsto_def)
apply clarsimp
apply (erule dup_boxE) (* \<box>\<box>(Init G \<longrightarrow> H) *)
apply merge_box
apply (clarsimp elim!: STL4E [temp_use])
apply (rule dup_dmdD)
apply (subgoal_tac "sigmaa \<Turnstile> \<diamond>Init G")
apply (erule DmdImpl2)
apply assumption
apply (simp add: dmdInitD)
done
lemma LatticeDisjunctionElim1: "\<turnstile> (F \<or> G \<leadsto> H) \<longrightarrow> (F \<leadsto> H)"
apply (unfold leadsto_def)
apply (auto simp: Init_simps elim!: STL4E [temp_use])
done
lemma LatticeDisjunctionElim2: "\<turnstile> (F \<or> G \<leadsto> H) \<longrightarrow> (G \<leadsto> H)"
apply (unfold leadsto_def)
apply (auto simp: Init_simps elim!: STL4E [temp_use])
done
lemma LatticeDisjunctionIntro: "\<turnstile> (F \<leadsto> H) \<and> (G \<leadsto> H) \<longrightarrow> (F \<or> G \<leadsto> H)"
apply (unfold leadsto_def)
apply clarsimp
apply merge_box
apply (auto simp: Init_simps elim!: STL4E [temp_use])
done
lemma LatticeDisjunction: "\<turnstile> (F \<or> G \<leadsto> H) = ((F \<leadsto> H) \<and> (G \<leadsto> H))"
by (auto intro: LatticeDisjunctionIntro [temp_use]
LatticeDisjunctionElim1 [temp_use]
LatticeDisjunctionElim2 [temp_use])
lemma LatticeDiamond: "\<turnstile> (A \<leadsto> B \<or> C) \<and> (B \<leadsto> D) \<and> (C \<leadsto> D) \<longrightarrow> (A \<leadsto> D)"
apply clarsimp
apply (subgoal_tac "sigma \<Turnstile> (B \<or> C) \<leadsto> D")
apply (erule_tac G = "LIFT (B \<or> C)" in LatticeTransitivity [temp_use])
apply (fastforce intro!: LatticeDisjunctionIntro [temp_use])+
done
lemma LatticeTriangle: "\<turnstile> (A \<leadsto> D \<or> B) \<and> (B \<leadsto> D) \<longrightarrow> (A \<leadsto> D)"
apply clarsimp
apply (subgoal_tac "sigma \<Turnstile> (D \<or> B) \<leadsto> D")
apply (erule_tac G = "LIFT (D \<or> B)" in LatticeTransitivity [temp_use])
apply assumption
apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
done
lemma LatticeTriangle2: "\<turnstile> (A \<leadsto> B \<or> D) \<and> (B \<leadsto> D) \<longrightarrow> (A \<leadsto> D)"
apply clarsimp
apply (subgoal_tac "sigma \<Turnstile> B \<or> D \<leadsto> D")
apply (erule_tac G = "LIFT (B \<or> 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:
"\<lbrakk> \<turnstile> $P \<and> N \<longrightarrow> P` \<or> Q`;
\<turnstile> ($P \<and> N) \<and> <A>_v \<longrightarrow> Q`;
\<turnstile> $P \<and> N \<longrightarrow> $(Enabled(<A>_v)) \<rbrakk>
\<Longrightarrow> \<turnstile> \<box>N \<and> WF(A)_v \<longrightarrow> (P \<leadsto> 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: "\<turnstile> N \<and> $P \<longrightarrow> $Enabled (<A>_v)"
and 2: "\<turnstile> N \<and> <A>_v \<longrightarrow> B"
and 3: "\<turnstile> \<box>(N \<and> [\<not>A]_v) \<longrightarrow> stable P"
shows "\<turnstile> \<box>N \<and> WF(A)_v \<longrightarrow> (P \<leadsto> 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:
"\<lbrakk> \<turnstile> $P \<and> N \<longrightarrow> P` \<or> Q`;
\<turnstile> ($P \<and> N) \<and> <A>_v \<longrightarrow> Q`;
\<turnstile> \<box>P \<and> \<box>N \<and> \<box>F \<longrightarrow> \<diamond>Enabled(<A>_v) \<rbrakk>
\<Longrightarrow> \<turnstile> \<box>N \<and> SF(A)_v \<and> \<box>F \<longrightarrow> (P \<leadsto> 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: "\<turnstile> N \<and> <B>_f \<longrightarrow> <M>_g"
and 2: "\<turnstile> $P \<and> P` \<and> <N \<and> A>_f \<longrightarrow> B"
and 3: "\<turnstile> P \<and> Enabled(<M>_g) \<longrightarrow> Enabled(<A>_f)"
and 4: "\<turnstile> \<box>(N \<and> [\<not>B]_f) \<and> WF(A)_f \<and> \<box>F \<and> \<diamond>\<box>Enabled(<M>_g) \<longrightarrow> \<diamond>\<box>P"
shows "\<turnstile> \<box>N \<and> WF(A)_f \<and> \<box>F \<longrightarrow> WF(M)_g"
apply (clarsimp dest!: BoxWFI [temp_use] BoxDmdBox [temp_use, THEN iffD2]
simp: WF_def [where A = M])
apply (erule_tac F = F in dup_boxE)
apply merge_temp_box
apply (erule STL4Edup)
apply assumption
apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
apply (rule classical)
apply (subgoal_tac "sigmaa \<Turnstile> \<diamond> (($P \<and> P` \<and> N) \<and> <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 \<Turnstile> \<diamond>\<box>P" in thin_rl)
apply (erule_tac V = "sigmaa \<Turnstile> \<box>F" in thin_rl)
apply (drule BoxWFI [temp_use])
apply (erule_tac F = "ACT N \<and> [\<not>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: "\<turnstile> N \<and> <B>_f \<longrightarrow> <M>_g"
and 2: "\<turnstile> $P \<and> P` \<and> <N \<and> A>_f \<longrightarrow> B"
and 3: "\<turnstile> P \<and> Enabled(<M>_g) \<longrightarrow> Enabled(<A>_f)"
and 4: "\<turnstile> \<box>(N \<and> [\<not>B]_f) \<and> SF(A)_f \<and> \<box>F \<and> \<box>\<diamond>Enabled(<M>_g) \<longrightarrow> \<diamond>\<box>P"
shows "\<turnstile> \<box>N \<and> SF(A)_f \<and> \<box>F \<longrightarrow> SF(M)_g"
apply (clarsimp dest!: BoxSFI [temp_use] simp: 2 [try_rewrite] SF_def [where A = M])
apply (erule_tac F = F in dup_boxE)
apply (erule_tac F = "TEMP \<diamond>Enabled (<M>_g) " in dup_boxE)
apply merge_temp_box
apply (erule STL4Edup)
apply assumption
apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
apply (rule classical)
apply (subgoal_tac "sigmaa \<Turnstile> \<diamond> (($P \<and> P` \<and> N) \<and> <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 \<Turnstile> \<box>F" in thin_rl)
apply (drule BoxSFI [temp_use])
apply (erule_tac F = "TEMP \<diamond>Enabled (<M>_g)" in dup_boxE)
apply (erule_tac F = "ACT N \<and> [\<not>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: "\<And>x. sigma \<Turnstile> F x \<leadsto> (G \<or> (\<exists>y. #((y,x)\<in>r) \<and> F y)) "
shows "sigma \<Turnstile> F x \<leadsto> 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) \<in> 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: "\<And>r. wf r \<Longrightarrow> \<turnstile> \<box>[ (v`, $v) \<in> #r ]_v \<longrightarrow> \<diamond>\<box>[#False]_v"
apply clarsimp
apply (rule ccontr)
apply (subgoal_tac "sigma \<Turnstile> (\<exists>x. v=#x) \<leadsto> #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 \<Longrightarrow> \<turnstile> \<diamond>\<box>[ (v`, $v) : #r ]_v \<longrightarrow> \<diamond>\<box>[#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 "\<turnstile> \<box>\<diamond>((v`, $v) \<in> #r) \<longrightarrow> \<box>\<diamond><(v`, $v) \<notin> #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 \<Turnstile> \<box>\<diamond> ((#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: "\<And>n::nat stfun. \<turnstile> \<box>\<diamond>(n` < $n) \<longrightarrow> \<box>\<diamond>($n < n`)"
apply clarsimp
apply (subgoal_tac "sigma \<Turnstile> \<box>\<diamond><\<not> ((n`,$n) \<in> #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: "(\<And>x. basevars (x,vs) \<Longrightarrow> sigma \<Turnstile> F x)"
shows "sigma \<Turnstile> (\<forall>\<forall>x. F x)"
by (auto simp: aall_def elim!: eexE [temp_use] intro!: 1 dest!: 2 [temp_use])
lemma aallE: "\<turnstile> (\<forall>\<forall>x. F x) \<longrightarrow> 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 \<Turnstile> \<exists>\<exists>x. F x"
and 2: "\<And>x. sigma \<Turnstile> F x \<longrightarrow> G x"
shows "sigma \<Turnstile> \<exists>\<exists>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 \<Turnstile> \<forall>\<forall>x. F(x)"
and 2: "\<And>x. sigma \<Turnstile> F(x) \<longrightarrow> G(x)"
shows "sigma \<Turnstile> \<forall>\<forall>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 \<Turnstile> Init I"
and 2: "sigma \<Turnstile> \<box>N"
and 3: "basevars vs"
and 4: "\<And>h. basevars(h,vs) \<Longrightarrow> \<turnstile> I \<and> h = ha \<longrightarrow> HI h"
and 5: "\<And>h s t. \<lbrakk> basevars(h,vs); N (s,t); h t = hb (h s) (s,t) \<rbrakk> \<Longrightarrow> HN h (s,t)"
shows "sigma \<Turnstile> \<exists>\<exists>h. Init (HI h) \<and> \<box>(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 "\<turnstile> \<exists>\<exists>h. Init(h = #True) \<and> \<box>(h` = (\<not>$h))"
apply (rule tempI)
apply (rule historyI)
apply (force simp: Init_defs intro!: unit_base [temp_use] necT [temp_use])+
done
end