(*  Title:      HOL/TLA/TLA.thy

     Author:     Stephan Merz

     Copyright:  1998 University of Munich

 *)

 header {* The temporal level of TLA *}

 theory TLA

 imports Init

 begin

 consts

   (** abstract syntax **)

   Box        :: "('w::world) form => temporal"

   Dmd        :: "('w::world) form => temporal"

   leadsto    :: "['w::world form, 'v::world form] => temporal"

   Stable     :: "stpred => temporal"

   WF         :: "[action, 'a stfun] => temporal"

   SF         :: "[action, 'a stfun] => temporal"

   (* Quantification over (flexible) state variables *)

   EEx        :: "('a stfun => temporal) => temporal"       (binder "Eex " 10)

   AAll       :: "('a stfun => temporal) => temporal"       (binder "Aall " 10)

   (** concrete syntax **)

 syntax

   "_Box"     :: "lift => lift"                        ("([]_)" [40] 40)

   "_Dmd"     :: "lift => lift"                        ("(<>_)" [40] 40)

   "_leadsto" :: "[lift,lift] => lift"                 ("(_ ~> _)" [23,22] 22)

   "_stable"  :: "lift => lift"                        ("(stable/ _)")

   "_WF"      :: "[lift,lift] => lift"                 ("(WF'(_')'_(_))" [0,60] 55)

   "_SF"      :: "[lift,lift] => lift"                 ("(SF'(_')'_(_))" [0,60] 55)

   "_EEx"     :: "[idts, lift] => lift"                ("(3EEX _./ _)" [0,10] 10)

   "_AAll"    :: "[idts, lift] => lift"                ("(3AALL _./ _)" [0,10] 10)

 translations

   "_Box"      ==   "CONST Box"

   "_Dmd"      ==   "CONST Dmd"

   "_leadsto"  ==   "CONST leadsto"

   "_stable"   ==   "CONST Stable"

   "_WF"       ==   "CONST WF"

   "_SF"       ==   "CONST SF"

   "_EEx v A"  ==   "Eex v. A"

   "_AAll v A" ==   "Aall v. A"

   "sigma |= []F"         <= "_Box F sigma"

   "sigma |= <>F"         <= "_Dmd F sigma"

   "sigma |= F ~> G"      <= "_leadsto F G sigma"

   "sigma |= stable P"    <= "_stable P sigma"

   "sigma |= WF(A)_v"     <= "_WF A v sigma"

   "sigma |= SF(A)_v"     <= "_SF A v sigma"

   "sigma |= EEX x. F"    <= "_EEx x F sigma"

   "sigma |= AALL x. F"    <= "_AAll x F sigma"

 syntax (xsymbols)

   "_Box"     :: "lift => lift"                        ("(\<box>_)" [40] 40)

   "_Dmd"     :: "lift => lift"                        ("(\<diamond>_)" [40] 40)

   "_leadsto" :: "[lift,lift] => lift"                 ("(_ \<leadsto> _)" [23,22] 22)

   "_EEx"     :: "[idts, lift] => lift"                ("(3\<exists>\<exists> _./ _)" [0,10] 10)

   "_AAll"    :: "[idts, lift] => lift"                ("(3\<forall>\<forall> _./ _)" [0,10] 10)

 syntax (HTML output)

   "_EEx"     :: "[idts, lift] => lift"                ("(3\<exists>\<exists> _./ _)" [0,10] 10)

   "_AAll"    :: "[idts, lift] => lift"                ("(3\<forall>\<forall> _./ _)" [0,10] 10)

 axiomatization where

   (* Definitions of derived operators *)

   dmd_def:      "\<And>F. TEMP <>F  ==  TEMP ~[]~F"

 axiomatization where

   boxInit:      "\<And>F. TEMP []F  ==  TEMP []Init F" and

   leadsto_def:  "\<And>F G. TEMP F ~> G  ==  TEMP [](Init F --> <>G)" and

   stable_def:   "\<And>P. TEMP stable P  ==  TEMP []($P --> P$)" and

   WF_def:       "TEMP WF(A)_v  ==  TEMP <>[] Enabled(<A>_v) --> []<><A>_v" and

   SF_def:       "TEMP SF(A)_v  ==  TEMP []<> Enabled(<A>_v) --> []<><A>_v" and

   aall_def:     "TEMP (AALL x. F x)  ==  TEMP ~ (EEX x. ~ F x)"

 axiomatization where

 (* Base axioms for raw TLA. *)

   normalT:    "\<And>F G. |- [](F --> G) --> ([]F --> []G)" and    (* polymorphic *)

   reflT:      "\<And>F. |- []F --> F" and         (* F::temporal *)

   transT:     "\<And>F. |- []F --> [][]F" and     (* polymorphic *)

   linT:       "\<And>F G. |- <>F & <>G --> (<>(F & <>G)) | (<>(G & <>F))" and

   discT:      "\<And>F. |- [](F --> <>(~F & <>F)) --> (F --> []<>F)" and

   primeI:     "\<And>P. |- []P --> Init P`" and

   primeE:     "\<And>P F. |- [](Init P --> []F) --> Init P` --> (F --> []F)" and

   indT:       "\<And>P F. |- [](Init P & ~[]F --> Init P` & F) --> Init P --> []F" and

   allT:       "\<And>F. |- (ALL x. [](F x)) = ([](ALL x. F x))"

 axiomatization where

   necT:       "\<And>F. |- F ==> |- []F"      (* polymorphic *)

 axiomatization where

 (* Flexible quantification: refinement mappings, history variables *)

   eexI:       "|- F x --> (EEX x. F x)" and

   eexE:       "[| sigma |= (EEX x. F x); basevars vs;

                  (!!x. [| basevars (x, vs); sigma |= F x |] ==> (G sigma)::bool)

               |] ==> G sigma" and

   history:    "|- EEX h. Init(h = ha) & [](!x. $h = #x --> h` = hb x)"

 (* Specialize intensional introduction/elimination rules for temporal formulas *)

 lemma tempI [intro!]: "(!!sigma. sigma |= (F::temporal)) ==> |- F"

   apply (rule intI)

   apply (erule meta_spec)

   done

 lemma tempD [dest]: "|- (F::temporal) ==> sigma |= F"

   by (erule intD)

 (* ======== Functions to "unlift" temporal theorems ====== *)

 ML {*

 (* The following functions are specialized versions of the corresponding

    functions defined in theory Intensional in that they introduce a

    "world" parameter of type "behavior".

 *)

 fun temp_unlift ctxt th =

   (rewrite_rule ctxt @{thms action_rews} (th RS @{thm tempD}))

     handle THM _ => action_unlift ctxt th;

 (* Turn  |- F = G  into meta-level rewrite rule  F == G *)

 val temp_rewrite = int_rewrite

 fun temp_use ctxt th =

   case (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;

 *}

 attribute_setup temp_unlift =

   {* Scan.succeed (Thm.rule_attribute (temp_unlift o Context.proof_of)) *}

 attribute_setup temp_rewrite =

   {* Scan.succeed (Thm.rule_attribute (temp_rewrite o Context.proof_of)) *}

 attribute_setup temp_use =

   {* Scan.succeed (Thm.rule_attribute (temp_use o Context.proof_of)) *}

 attribute_setup try_rewrite =

   {* Scan.succeed (Thm.rule_attribute (try_rewrite o Context.proof_of)) *}

 (* ------------------------------------------------------------------------- *)

 (***           "Simple temporal logic": only [] and <>                     ***)

 (* ------------------------------------------------------------------------- *)

 section "Simple temporal logic"

 (* []~F == []~Init F *)

 lemmas boxNotInit = boxInit [of "LIFT ~F", unfolded Init_simps] for F

 lemma dmdInit: "TEMP <>F == TEMP <> Init F"

   apply (unfold dmd_def)

   apply (unfold boxInit [of "LIFT ~F"])

   apply (simp (no_asm) add: Init_simps)

   done

 lemmas dmdNotInit = dmdInit [of "LIFT ~F", unfolded Init_simps] for F

 (* boxInit and dmdInit cannot be used as rewrites, because they loop.

    Non-looping instances for state predicates and actions are occasionally useful.

 *)

 lemmas boxInit_stp = boxInit [where 'a = state]

 lemmas boxInit_act = boxInit [where 'a = "state * state"]

 lemmas dmdInit_stp = dmdInit [where 'a = state]

 lemmas dmdInit_act = dmdInit [where 'a = "state * state"]

 (* The symmetric equations can be used to get rid of Init *)

 lemmas boxInitD = boxInit [symmetric]

 lemmas dmdInitD = dmdInit [symmetric]

 lemmas boxNotInitD = boxNotInit [symmetric]

 lemmas dmdNotInitD = dmdNotInit [symmetric]

 lemmas Init_simps = Init_simps boxInitD dmdInitD boxNotInitD dmdNotInitD

 (* ------------------------ STL2 ------------------------------------------- *)

 lemmas STL2 = reflT

 (* The "polymorphic" (generic) variant *)

 lemma STL2_gen: "|- []F --> Init F"

   apply (unfold boxInit [of F])

   apply (rule STL2)

   done

 (* see also STL2_pr below: "|- []P --> Init P & Init (P`)" *)

 (* Dual versions for <> *)

 lemma InitDmd: "|- F --> <> F"

   apply (unfold dmd_def)

   apply (auto dest!: STL2 [temp_use])

   done

 lemma InitDmd_gen: "|- Init F --> <>F"

   apply clarsimp

   apply (drule InitDmd [temp_use])

   apply (simp add: dmdInitD)

   done

 (* ------------------------ STL3 ------------------------------------------- *)

 lemma STL3: "|- ([][]F) = ([]F)"

   by (auto elim: transT [temp_use] STL2 [temp_use])

 (* corresponding elimination rule introduces double boxes:

    [| (sigma |= []F); (sigma |= [][]F) ==> PROP W |] ==> PROP W

 *)

 lemmas dup_boxE = STL3 [temp_unlift, THEN iffD2, elim_format]

 lemmas dup_boxD = STL3 [temp_unlift, THEN iffD1]

 (* dual versions for <> *)

 lemma DmdDmd: "|- (<><>F) = (<>F)"

   by (auto simp add: dmd_def [try_rewrite] STL3 [try_rewrite])

 lemmas dup_dmdE = DmdDmd [temp_unlift, THEN iffD2, elim_format]

 lemmas dup_dmdD = DmdDmd [temp_unlift, THEN iffD1]

 (* ------------------------ STL4 ------------------------------------------- *)

 lemma STL4:

   assumes "|- F --> G"

   shows "|- []F --> []G"

   apply clarsimp

   apply (rule normalT [temp_use])

    apply (rule assms [THEN necT, temp_use])

   apply assumption

   done

 (* Unlifted version as an elimination rule *)

 lemma STL4E: "[| sigma |= []F; |- F --> G |] ==> sigma |= []G"

   by (erule (1) STL4 [temp_use])

 lemma STL4_gen: "|- Init F --> Init G ==> |- []F --> []G"

   apply (drule STL4)

   apply (simp add: boxInitD)

   done

 lemma STL4E_gen: "[| sigma |= []F; |- Init F --> Init G |] ==> sigma |= []G"

   by (erule (1) STL4_gen [temp_use])

 (* see also STL4Edup below, which allows an auxiliary boxed formula:

        []A /\ F => G

      -----------------

      []A /\ []F => []G

 *)

 (* The dual versions for <> *)

 lemma DmdImpl:

   assumes prem: "|- F --> G"

   shows "|- <>F --> <>G"

   apply (unfold dmd_def)

   apply (fastforce intro!: prem [temp_use] elim!: STL4E [temp_use])

   done

 lemma DmdImplE: "[| sigma |= <>F; |- F --> G |] ==> sigma |= <>G"

   by (erule (1) DmdImpl [temp_use])

 (* ------------------------ STL5 ------------------------------------------- *)

 lemma STL5: "|- ([]F & []G) = ([](F & G))"

   apply auto

   apply (subgoal_tac "sigma |= [] (G --> (F & G))")

      apply (erule normalT [temp_use])

      apply (fastforce elim!: STL4E [temp_use])+

   done

 (* rewrite rule to split conjunctions under boxes *)

 lemmas split_box_conj = STL5 [temp_unlift, symmetric]

 (* the corresponding elimination rule allows to combine boxes in the hypotheses

    (NB: F and G must have the same type, i.e., both actions or temporals.)

    Use "addSE2" etc. if you want to add this to a claset, otherwise it will loop!

 *)

 lemma box_conjE:

   assumes "sigma |= []F"

      and "sigma |= []G"

   and "sigma |= [](F&G) ==> PROP R"

   shows "PROP R"

   by (rule assms STL5 [temp_unlift, THEN iffD1] conjI)+

 (* Instances of box_conjE for state predicates, actions, and temporals

    in case the general rule is "too polymorphic".

 *)

 lemmas box_conjE_temp = box_conjE [where 'a = behavior]

 lemmas box_conjE_stp = box_conjE [where 'a = state]

 lemmas box_conjE_act = box_conjE [where 'a = "state * state"]

 (* Define a tactic that tries to merge all boxes in an antecedent. The definition is

    a bit kludgy in order to simulate "double elim-resolution".

 *)

 lemma box_thin: "[| sigma |= []F; PROP W |] ==> PROP W" .

 ML {*

 fun merge_box_tac i =

    REPEAT_DETERM (EVERY [etac @{thm box_conjE} i, atac i, etac @{thm box_thin} i])

 fun merge_temp_box_tac ctxt i =

    REPEAT_DETERM (EVERY [etac @{thm box_conjE_temp} i, atac i,

                          eres_inst_tac ctxt [(("'a", 0), "behavior")] @{thm box_thin} i])

 fun merge_stp_box_tac ctxt i =

    REPEAT_DETERM (EVERY [etac @{thm box_conjE_stp} i, atac i,

                          eres_inst_tac ctxt [(("'a", 0), "state")] @{thm box_thin} i])

 fun merge_act_box_tac ctxt i =

    REPEAT_DETERM (EVERY [etac @{thm box_conjE_act} i, atac i,

                          eres_inst_tac ctxt [(("'a", 0), "state * state")] @{thm box_thin} i])

 *}

 method_setup merge_box = {* Scan.succeed (K (SIMPLE_METHOD' merge_box_tac)) *}

 method_setup merge_temp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_temp_box_tac) *}

 method_setup merge_stp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_stp_box_tac) *}

 method_setup merge_act_box = {* Scan.succeed (SIMPLE_METHOD' o merge_act_box_tac) *}

 (* rewrite rule to push universal quantification through box:

       (sigma |= [](! x. F x)) = (! x. (sigma |= []F x))

 *)

 lemmas all_box = allT [temp_unlift, symmetric]

 lemma DmdOr: "|- (<>(F | G)) = (<>F | <>G)"

   apply (auto simp add: dmd_def split_box_conj [try_rewrite])

   apply (erule contrapos_np, merge_box, fastforce elim!: STL4E [temp_use])+

   done

 lemma exT: "|- (EX x. <>(F x)) = (<>(EX x. F x))"

   by (auto simp: dmd_def Not_Rex [try_rewrite] all_box [try_rewrite])

 lemmas ex_dmd = exT [temp_unlift, symmetric]

 lemma STL4Edup: "!!sigma. [| sigma |= []A; sigma |= []F; |- F & []A --> G |] ==> sigma |= []G"

   apply (erule dup_boxE)

   apply merge_box

   apply (erule STL4E)

   apply assumption

   done

 lemma DmdImpl2: 

     "!!sigma. [| sigma |= <>F; sigma |= [](F --> G) |] ==> sigma |= <>G"

   apply (unfold dmd_def)

   apply auto

   apply (erule notE)

   apply merge_box

   apply (fastforce elim!: STL4E [temp_use])

   done

 lemma InfImpl:

   assumes 1: "sigma |= []<>F"

     and 2: "sigma |= []G"

     and 3: "|- F & G --> H"

   shows "sigma |= []<>H"

   apply (insert 1 2)

   apply (erule_tac F = G in dup_boxE)

   apply merge_box

   apply (fastforce elim!: STL4E [temp_use] DmdImpl2 [temp_use] intro!: 3 [temp_use])

   done

 (* ------------------------ STL6 ------------------------------------------- *)

 (* Used in the proof of STL6, but useful in itself. *)

 lemma BoxDmd: "|- []F & <>G --> <>([]F & G)"

   apply (unfold dmd_def)

   apply clarsimp

   apply (erule dup_boxE)

   apply merge_box

   apply (erule contrapos_np)

   apply (fastforce elim!: STL4E [temp_use])

   done

 (* weaker than BoxDmd, but more polymorphic (and often just right) *)

 lemma BoxDmd_simple: "|- []F & <>G --> <>(F & G)"

   apply (unfold dmd_def)

   apply clarsimp

   apply merge_box

   apply (fastforce elim!: notE STL4E [temp_use])

   done

 lemma BoxDmd2_simple: "|- []F & <>G --> <>(G & F)"

   apply (unfold dmd_def)

   apply clarsimp

   apply merge_box

   apply (fastforce elim!: notE STL4E [temp_use])

   done

 lemma DmdImpldup:

   assumes 1: "sigma |= []A"

     and 2: "sigma |= <>F"

     and 3: "|- []A & F --> G"

   shows "sigma |= <>G"

   apply (rule 2 [THEN 1 [THEN BoxDmd [temp_use]], THEN DmdImplE])

   apply (rule 3)

   done

 lemma STL6: "|- <>[]F & <>[]G --> <>[](F & G)"

   apply (auto simp: STL5 [temp_rewrite, symmetric])

   apply (drule linT [temp_use])

    apply assumption

   apply (erule thin_rl)

   apply (rule DmdDmd [temp_unlift, THEN iffD1])

   apply (erule disjE)

    apply (erule DmdImplE)

    apply (rule BoxDmd)

   apply (erule DmdImplE)

   apply auto

   apply (drule BoxDmd [temp_use])

    apply assumption

   apply (erule thin_rl)

   apply (fastforce elim!: DmdImplE [temp_use])

   done

 (* ------------------------ True / False ----------------------------------------- *)

 section "Simplification of constants"

 lemma BoxConst: "|- ([]#P) = #P"

   apply (rule tempI)

   apply (cases P)

    apply (auto intro!: necT [temp_use] dest: STL2_gen [temp_use] simp: Init_simps)

   done

 lemma DmdConst: "|- (<>#P) = #P"

   apply (unfold dmd_def)

   apply (cases P)

   apply (simp_all add: BoxConst [try_rewrite])

   done

 lemmas temp_simps [temp_rewrite, simp] = BoxConst DmdConst

 (* ------------------------ Further rewrites ----------------------------------------- *)

 section "Further rewrites"

 lemma NotBox: "|- (~[]F) = (<>~F)"

   by (simp add: dmd_def)

 lemma NotDmd: "|- (~<>F) = ([]~F)"

   by (simp add: dmd_def)

 (* These are not declared by default, because they could be harmful,

    e.g. []F & ~[]F becomes []F & <>~F !! *)

 lemmas more_temp_simps1 =

   STL3 [temp_rewrite] DmdDmd [temp_rewrite] NotBox [temp_rewrite] NotDmd [temp_rewrite]

   NotBox [temp_unlift, THEN eq_reflection]

   NotDmd [temp_unlift, THEN eq_reflection]

 lemma BoxDmdBox: "|- ([]<>[]F) = (<>[]F)"

   apply (auto dest!: STL2 [temp_use])

   apply (rule ccontr)

   apply (subgoal_tac "sigma |= <>[][]F & <>[]~[]F")

    apply (erule thin_rl)

    apply auto

     apply (drule STL6 [temp_use])

      apply assumption

     apply simp

    apply (simp_all add: more_temp_simps1)

   done

 lemma DmdBoxDmd: "|- (<>[]<>F) = ([]<>F)"

   apply (unfold dmd_def)

   apply (auto simp: BoxDmdBox [unfolded dmd_def, try_rewrite])

   done

 lemmas more_temp_simps2 = more_temp_simps1 BoxDmdBox [temp_rewrite] DmdBoxDmd [temp_rewrite]

 (* ------------------------ Miscellaneous ----------------------------------- *)

 lemma BoxOr: "!!sigma. [| sigma |= []F | []G |] ==> sigma |= [](F | G)"

   by (fastforce elim!: STL4E [temp_use])

 (* "persistently implies infinitely often" *)

 lemma DBImplBD: "|- <>[]F --> []<>F"

   apply clarsimp

   apply (rule ccontr)

   apply (simp add: more_temp_simps2)

   apply (drule STL6 [temp_use])

    apply assumption

   apply simp

   done

 lemma BoxDmdDmdBox: "|- []<>F & <>[]G --> []<>(F & G)"

   apply clarsimp

   apply (rule ccontr)

   apply (unfold more_temp_simps2)

   apply (drule STL6 [temp_use])

    apply assumption

   apply (subgoal_tac "sigma |= <>[]~F")

    apply (force simp: dmd_def)

   apply (fastforce elim: DmdImplE [temp_use] STL4E [temp_use])

   done

 (* ------------------------------------------------------------------------- *)

 (***          TLA-specific theorems: primed formulas                       ***)

 (* ------------------------------------------------------------------------- *)

 section "priming"

 (* ------------------------ TLA2 ------------------------------------------- *)

 lemma STL2_pr: "|- []P --> Init P & Init P`"

   by (fastforce intro!: STL2_gen [temp_use] primeI [temp_use])

 (* Auxiliary lemma allows priming of boxed actions *)

 lemma BoxPrime: "|- []P --> []($P & P$)"

   apply clarsimp

   apply (erule dup_boxE)

   apply (unfold boxInit_act)

   apply (erule STL4E)

   apply (auto simp: Init_simps dest!: STL2_pr [temp_use])

   done

 lemma TLA2:

   assumes "|- $P & P$ --> A"

   shows "|- []P --> []A"

   apply clarsimp

   apply (drule BoxPrime [temp_use])

   apply (auto simp: Init_stp_act_rev [try_rewrite] intro!: assms [temp_use]

     elim!: STL4E [temp_use])

   done

 lemma TLA2E: "[| sigma |= []P; |- $P & P$ --> A |] ==> sigma |= []A"

   by (erule (1) TLA2 [temp_use])

 lemma DmdPrime: "|- (<>P`) --> (<>P)"

   apply (unfold dmd_def)

   apply (fastforce elim!: TLA2E [temp_use])

   done

 lemmas PrimeDmd = InitDmd_gen [temp_use, THEN DmdPrime [temp_use]]

 (* ------------------------ INV1, stable --------------------------------------- *)

 section "stable, invariant"

 lemma ind_rule:

    "[| sigma |= []H; sigma |= Init P; |- H --> (Init P & ~[]F --> Init(P`) & F) |]  

     ==> sigma |= []F"

   apply (rule indT [temp_use])

    apply (erule (2) STL4E)

   done

 lemma box_stp_act: "|- ([]$P) = ([]P)"

   by (simp add: boxInit_act Init_simps)

 lemmas box_stp_actI = box_stp_act [temp_use, THEN iffD2]

 lemmas box_stp_actD = box_stp_act [temp_use, THEN iffD1]

 lemmas more_temp_simps3 = box_stp_act [temp_rewrite] more_temp_simps2

 lemma INV1: 

   "|- (Init P) --> (stable P) --> []P"

   apply (unfold stable_def boxInit_stp boxInit_act)

   apply clarsimp

   apply (erule ind_rule)

    apply (auto simp: Init_simps elim: ind_rule)

   done

 lemma StableT: 

     "!!P. |- $P & A --> P` ==> |- []A --> stable P"

   apply (unfold stable_def)

   apply (fastforce elim!: STL4E [temp_use])

   done

 lemma Stable: "[| sigma |= []A; |- $P & A --> P` |] ==> sigma |= stable P"

   by (erule (1) StableT [temp_use])

 (* Generalization of INV1 *)

 lemma StableBox: "|- (stable P) --> [](Init P --> []P)"

   apply (unfold stable_def)

   apply clarsimp

   apply (erule dup_boxE)

   apply (force simp: stable_def elim: STL4E [temp_use] INV1 [temp_use])

   done

 lemma DmdStable: "|- (stable P) & <>P --> <>[]P"

   apply clarsimp

   apply (rule DmdImpl2)

    prefer 2

    apply (erule StableBox [temp_use])

   apply (simp add: dmdInitD)

   done

 (* ---------------- (Semi-)automatic invariant tactics ---------------------- *)

 ML {*

 (* inv_tac reduces goals of the form ... ==> sigma |= []P *)

 fun inv_tac ctxt =

   SELECT_GOAL

     (EVERY

      [auto_tac ctxt,

       TRY (merge_box_tac 1),

       rtac (temp_use ctxt @{thm INV1}) 1, (* fail if the goal is not a box *)

       TRYALL (etac @{thm Stable})]);

 (* auto_inv_tac applies inv_tac and then tries to attack the subgoals

    in simple cases it may be able to handle goals like |- MyProg --> []Inv.

    In these simple cases the simplifier seems to be more useful than the

    auto-tactic, which applies too much propositional logic and simplifies

    too late.

 *)

 fun auto_inv_tac ctxt =

   SELECT_GOAL

     (inv_tac ctxt 1 THEN

       (TRYALL (action_simp_tac

         (ctxt addsimps [@{thm Init_stp}, @{thm Init_act}]) [] [@{thm squareE}])));

 *}

 method_setup invariant = {*

   Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o inv_tac))

 *}

 method_setup auto_invariant = {*

   Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o auto_inv_tac))

 *}

 lemma unless: "|- []($P --> P` | Q`) --> (stable P) | <>Q"

   apply (unfold dmd_def)

   apply (clarsimp dest!: BoxPrime [temp_use])

   apply merge_box

   apply (erule contrapos_np)

   apply (fastforce elim!: Stable [temp_use])

   done

 (* --------------------- Recursive expansions --------------------------------------- *)

 section "recursive expansions"

 (* Recursive expansions of [] and <> for state predicates *)

 lemma BoxRec: "|- ([]P) = (Init P & []P`)"

   apply (auto intro!: STL2_gen [temp_use])

    apply (fastforce elim!: TLA2E [temp_use])

   apply (auto simp: stable_def elim!: INV1 [temp_use] STL4E [temp_use])

   done

 lemma DmdRec: "|- (<>P) = (Init P | <>P`)"

   apply (unfold dmd_def BoxRec [temp_rewrite])

   apply (auto simp: Init_simps)

   done

 lemma DmdRec2: "!!sigma. [| sigma |= <>P; sigma |= []~P` |] ==> sigma |= Init P"

   apply (force simp: DmdRec [temp_rewrite] dmd_def)

   done

 lemma InfinitePrime: "|- ([]<>P) = ([]<>P`)"

   apply auto

    apply (rule classical)

    apply (rule DBImplBD [temp_use])

    apply (subgoal_tac "sigma |= <>[]P")

     apply (fastforce elim!: DmdImplE [temp_use] TLA2E [temp_use])

    apply (subgoal_tac "sigma |= <>[] (<>P & []~P`)")

     apply (force simp: boxInit_stp [temp_use]

       elim!: DmdImplE [temp_use] STL4E [temp_use] DmdRec2 [temp_use])

    apply (force intro!: STL6 [temp_use] simp: more_temp_simps3)

   apply (fastforce intro: DmdPrime [temp_use] elim!: STL4E [temp_use])

   done

 lemma InfiniteEnsures:

   "[| sigma |= []N; sigma |= []<>A; |- A & N --> P` |] ==> sigma |= []<>P"

   apply (unfold InfinitePrime [temp_rewrite])

   apply (rule InfImpl)

     apply assumption+

   done

 (* ------------------------ fairness ------------------------------------------- *)

 section "fairness"

 (* alternative definitions of fairness *)

 lemma WF_alt: "|- WF(A)_v = ([]<>~Enabled(<A>_v) | []<><A>_v)"

   apply (unfold WF_def dmd_def)

   apply fastforce

   done

 lemma SF_alt: "|- SF(A)_v = (<>[]~Enabled(<A>_v) | []<><A>_v)"

   apply (unfold SF_def dmd_def)

   apply fastforce

   done

 (* theorems to "box" fairness conditions *)

 lemma BoxWFI: "|- WF(A)_v --> []WF(A)_v"

   by (auto simp: WF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])

 lemma WF_Box: "|- ([]WF(A)_v) = WF(A)_v"

   by (fastforce intro!: BoxWFI [temp_use] dest!: STL2 [temp_use])

 lemma BoxSFI: "|- SF(A)_v --> []SF(A)_v"

   by (auto simp: SF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])

 lemma SF_Box: "|- ([]SF(A)_v) = SF(A)_v"

   by (fastforce intro!: BoxSFI [temp_use] dest!: STL2 [temp_use])

 lemmas more_temp_simps = more_temp_simps3 WF_Box [temp_rewrite] SF_Box [temp_rewrite]

 lemma SFImplWF: "|- SF(A)_v --> WF(A)_v"

   apply (unfold SF_def WF_def)

   apply (fastforce dest!: DBImplBD [temp_use])

   done

 (* A tactic that "boxes" all fairness conditions. Apply more_temp_simps to "unbox". *)

 ML {*

 val box_fair_tac = SELECT_GOAL (REPEAT (dresolve_tac [@{thm BoxWFI}, @{thm BoxSFI}] 1))

 *}

 (* ------------------------------ leads-to ------------------------------ *)

 section "~>"

 lemma leadsto_init: "|- (Init F) & (F ~> G) --> <>G"

   apply (unfold leadsto_def)

   apply (auto dest!: STL2 [temp_use])

   done

 (* |- F & (F ~> G) --> <>G *)

 lemmas leadsto_init_temp = leadsto_init [where 'a = behavior, unfolded Init_simps]

 lemma streett_leadsto: "|- ([]<>Init F --> []<>G) = (<>(F ~> G))"

   apply (unfold leadsto_def)

   apply auto

     apply (simp add: more_temp_simps)

     apply (fastforce elim!: DmdImplE [temp_use] STL4E [temp_use])

    apply (fastforce intro!: InitDmd [temp_use] elim!: STL4E [temp_use])

   apply (subgoal_tac "sigma |= []<><>G")

    apply (simp add: more_temp_simps)

   apply (drule BoxDmdDmdBox [temp_use])

    apply assumption

   apply (fastforce elim!: DmdImplE [temp_use] STL4E [temp_use])

   done

 lemma leadsto_infinite: "|- []<>F & (F ~> G) --> []<>G"

   apply clarsimp

   apply (erule InitDmd [temp_use, THEN streett_leadsto [temp_unlift, THEN iffD2, THEN mp]])

   apply (simp add: dmdInitD)

   done

 (* In particular, strong fairness is a Streett condition. The following

    rules are sometimes easier to use than WF2 or SF2 below.

 *)

 lemma leadsto_SF: "|- (Enabled(<A>_v) ~> <A>_v) --> SF(A)_v"

   apply (unfold SF_def)

   apply (clarsimp elim!: leadsto_infinite [temp_use])

   done

 lemma leadsto_WF: "|- (Enabled(<A>_v) ~> <A>_v) --> WF(A)_v"

   by (clarsimp intro!: SFImplWF [temp_use] leadsto_SF [temp_use])

 (* introduce an invariant into the proof of a leadsto assertion.

    []I --> ((P ~> Q)  =  (P /\ I ~> Q))

 *)

 lemma INV_leadsto: "|- []I & (P & I ~> Q) --> (P ~> Q)"

   apply (unfold leadsto_def)

   apply clarsimp

   apply (erule STL4Edup)

    apply assumption

   apply (auto simp: Init_simps dest!: STL2_gen [temp_use])

   done

 lemma leadsto_classical: "|- (Init F & []~G ~> G) --> (F ~> G)"

   apply (unfold leadsto_def dmd_def)

   apply (force simp: Init_simps elim!: STL4E [temp_use])

   done

 lemma leadsto_false: "|- (F ~> #False) = ([]~F)"

   apply (unfold leadsto_def)

   apply (simp add: boxNotInitD)

   done

 lemma leadsto_exists: "|- ((EX x. F x) ~> G) = (ALL x. (F x ~> G))"

   apply (unfold leadsto_def)

   apply (auto simp: allT [try_rewrite] Init_simps elim!: STL4E [temp_use])

   done

 (* basic leadsto properties, cf. Unity *)

 lemma ImplLeadsto_gen: "|- [](Init F --> Init G) --> (F ~> G)"

   apply (unfold leadsto_def)

   apply (auto intro!: InitDmd_gen [temp_use]

     elim!: STL4E_gen [temp_use] simp: Init_simps)

   done

 lemmas ImplLeadsto =

   ImplLeadsto_gen [where 'a = behavior and 'b = behavior, unfolded Init_simps]

 lemma ImplLeadsto_simple: "!!F G. |- F --> G ==> |- F ~> G"

   by (auto simp: Init_def intro!: ImplLeadsto_gen [temp_use] necT [temp_use])

 lemma EnsuresLeadsto:

   assumes "|- A & $P --> Q`"

   shows "|- []A --> (P ~> Q)"

   apply (unfold leadsto_def)

   apply (clarsimp elim!: INV_leadsto [temp_use])

   apply (erule STL4E_gen)

   apply (auto simp: Init_defs intro!: PrimeDmd [temp_use] assms [temp_use])

   done

 lemma EnsuresLeadsto2: "|- []($P --> Q`) --> (P ~> Q)"

   apply (unfold leadsto_def)

   apply clarsimp

   apply (erule STL4E_gen)

   apply (auto simp: Init_simps intro!: PrimeDmd [temp_use])

   done

 lemma ensures:

   assumes 1: "|- $P & N --> P` | Q`"

     and 2: "|- ($P & N) & A --> Q`"

   shows "|- []N & []([]P --> <>A) --> (P ~> Q)"

   apply (unfold leadsto_def)

   apply clarsimp

   apply (erule STL4Edup)

    apply assumption

   apply clarsimp

   apply (subgoal_tac "sigmaa |= [] ($P --> P` | Q`) ")

    apply (drule unless [temp_use])

    apply (clarsimp dest!: INV1 [temp_use])

   apply (rule 2 [THEN DmdImpl, temp_use, THEN DmdPrime [temp_use]])

    apply (force intro!: BoxDmd_simple [temp_use]

      simp: split_box_conj [try_rewrite] box_stp_act [try_rewrite])

   apply (force elim: STL4E [temp_use] dest: 1 [temp_use])

   done

 lemma ensures_simple:

   "[| |- $P & N --> P` | Q`;  

       |- ($P & N) & A --> Q`  

    |] ==> |- []N & []<>A --> (P ~> Q)"

   apply clarsimp

   apply (erule (2) ensures [temp_use])

   apply (force elim!: STL4E [temp_use])

   done

 lemma EnsuresInfinite:

     "[| sigma |= []<>P; sigma |= []A; |- A & $P --> Q` |] ==> sigma |= []<>Q"

   apply (erule leadsto_infinite [temp_use])

   apply (erule EnsuresLeadsto [temp_use])

   apply assumption

   done

 (*** Gronning's lattice rules (taken from TLP) ***)

 section "Lattice rules"

 lemma LatticeReflexivity: "|- F ~> F"

   apply (unfold leadsto_def)

   apply (rule necT InitDmd_gen)+

   done

 lemma LatticeTransitivity: "|- (G ~> H) & (F ~> G) --> (F ~> H)"

   apply (unfold leadsto_def)

   apply clarsimp

   apply (erule dup_boxE) (* [][] (Init G --> H) *)

   apply merge_box

   apply (clarsimp elim!: STL4E [temp_use])

   apply (rule dup_dmdD)

   apply (subgoal_tac "sigmaa |= <>Init G")

    apply (erule DmdImpl2)

    apply assumption

   apply (simp add: dmdInitD)

   done

 lemma LatticeDisjunctionElim1: "|- (F | G ~> H) --> (F ~> H)"

   apply (unfold leadsto_def)

   apply (auto simp: Init_simps elim!: STL4E [temp_use])

   done

 lemma LatticeDisjunctionElim2: "|- (F | G ~> H) --> (G ~> H)"

   apply (unfold leadsto_def)

   apply (auto simp: Init_simps elim!: STL4E [temp_use])

   done

 lemma LatticeDisjunctionIntro: "|- (F ~> H) & (G ~> H) --> (F | G ~> H)"

   apply (unfold leadsto_def)

   apply clarsimp

   apply merge_box

   apply (auto simp: Init_simps elim!: STL4E [temp_use])

   done

 lemma LatticeDisjunction: "|- (F | G ~> H) = ((F ~> H) & (G ~> H))"

   by (auto intro: LatticeDisjunctionIntro [temp_use]

     LatticeDisjunctionElim1 [temp_use]

     LatticeDisjunctionElim2 [temp_use])

 lemma LatticeDiamond: "|- (A ~> B | C) & (B ~> D) & (C ~> D) --> (A ~> D)"

   apply clarsimp

   apply (subgoal_tac "sigma |= (B | C) ~> D")

   apply (erule_tac G = "LIFT (B | C)" in LatticeTransitivity [temp_use])

    apply (fastforce intro!: LatticeDisjunctionIntro [temp_use])+

   done

 lemma LatticeTriangle: "|- (A ~> D | B) & (B ~> D) --> (A ~> D)"

   apply clarsimp

   apply (subgoal_tac "sigma |= (D | B) ~> D")

    apply (erule_tac G = "LIFT (D | B)" in LatticeTransitivity [temp_use])

   apply assumption

   apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])

   done

 lemma LatticeTriangle2: "|- (A ~> B | D) & (B ~> D) --> (A ~> D)"

   apply clarsimp

   apply (subgoal_tac "sigma |= B | D ~> D")

    apply (erule_tac G = "LIFT (B | D)" in LatticeTransitivity [temp_use])

    apply assumption

   apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])

   done

 (*** Lamport's fairness rules ***)

 section "Fairness rules"

 lemma WF1:

   "[| |- $P & N  --> P` | Q`;    

       |- ($P & N) & <A>_v --> Q`;    

       |- $P & N --> $(Enabled(<A>_v)) |]    

   ==> |- []N & WF(A)_v --> (P ~> Q)"

   apply (clarsimp dest!: BoxWFI [temp_use])

   apply (erule (2) ensures [temp_use])

   apply (erule (1) STL4Edup)

   apply (clarsimp simp: WF_def)

   apply (rule STL2 [temp_use])

   apply (clarsimp elim!: mp intro!: InitDmd [temp_use])

   apply (erule STL4 [temp_use, THEN box_stp_actD [temp_use]])

   apply (simp add: split_box_conj box_stp_actI)

   done

 (* Sometimes easier to use; designed for action B rather than state predicate Q *)

 lemma WF_leadsto:

   assumes 1: "|- N & $P --> $Enabled (<A>_v)"

     and 2: "|- N & <A>_v --> B"

     and 3: "|- [](N & [~A]_v) --> stable P"

   shows "|- []N & WF(A)_v --> (P ~> B)"

   apply (unfold leadsto_def)

   apply (clarsimp dest!: BoxWFI [temp_use])

   apply (erule (1) STL4Edup)

   apply clarsimp

   apply (rule 2 [THEN DmdImpl, temp_use])

   apply (rule BoxDmd_simple [temp_use])

    apply assumption

   apply (rule classical)

   apply (rule STL2 [temp_use])

   apply (clarsimp simp: WF_def elim!: mp intro!: InitDmd [temp_use])

   apply (rule 1 [THEN STL4, temp_use, THEN box_stp_actD])

   apply (simp (no_asm_simp) add: split_box_conj [try_rewrite] box_stp_act [try_rewrite])

   apply (erule INV1 [temp_use])

   apply (rule 3 [temp_use])

   apply (simp add: split_box_conj [try_rewrite] NotDmd [temp_use] not_angle [try_rewrite])

   done

 lemma SF1:

   "[| |- $P & N  --> P` | Q`;    

       |- ($P & N) & <A>_v --> Q`;    

       |- []P & []N & []F --> <>Enabled(<A>_v) |]    

   ==> |- []N & SF(A)_v & []F --> (P ~> Q)"

   apply (clarsimp dest!: BoxSFI [temp_use])

   apply (erule (2) ensures [temp_use])

   apply (erule_tac F = F in dup_boxE)

   apply merge_temp_box

   apply (erule STL4Edup)

   apply assumption

   apply (clarsimp simp: SF_def)

   apply (rule STL2 [temp_use])

   apply (erule mp)

   apply (erule STL4 [temp_use])

   apply (simp add: split_box_conj [try_rewrite] STL3 [try_rewrite])

   done

 lemma WF2:

   assumes 1: "|- N & <B>_f --> <M>_g"

     and 2: "|- $P & P` & <N & A>_f --> B"

     and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"

     and 4: "|- [](N & [~B]_f) & WF(A)_f & []F & <>[]Enabled(<M>_g) --> <>[]P"

   shows "|- []N & WF(A)_f & []F --> WF(M)_g"

   apply (clarsimp dest!: BoxWFI [temp_use] BoxDmdBox [temp_use, THEN iffD2]

     simp: WF_def [where A = M])

   apply (erule_tac F = F in dup_boxE)

   apply merge_temp_box

   apply (erule STL4Edup)

    apply assumption

   apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])

   apply (rule classical)

   apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")

    apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])

   apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])

   apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])

   apply merge_act_box

   apply (frule 4 [temp_use])

      apply assumption+

   apply (drule STL6 [temp_use])

    apply assumption

   apply (erule_tac V = "sigmaa |= <>[]P" in thin_rl)

   apply (erule_tac V = "sigmaa |= []F" in thin_rl)

   apply (drule BoxWFI [temp_use])

   apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)

   apply merge_temp_box

   apply (erule DmdImpldup)

    apply assumption

   apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]

     WF_Box [try_rewrite] box_stp_act [try_rewrite])

    apply (force elim!: TLA2E [where P = P, temp_use])

   apply (rule STL2 [temp_use])

   apply (force simp: WF_def split_box_conj [try_rewrite]

     elim!: mp intro!: InitDmd [temp_use] 3 [THEN STL4, temp_use])

   done

 lemma SF2:

   assumes 1: "|- N & <B>_f --> <M>_g"

     and 2: "|- $P & P` & <N & A>_f --> B"

     and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"

     and 4: "|- [](N & [~B]_f) & SF(A)_f & []F & []<>Enabled(<M>_g) --> <>[]P"

   shows "|- []N & SF(A)_f & []F --> SF(M)_g"

   apply (clarsimp dest!: BoxSFI [temp_use] simp: 2 [try_rewrite] SF_def [where A = M])

   apply (erule_tac F = F in dup_boxE)

   apply (erule_tac F = "TEMP <>Enabled (<M>_g) " in dup_boxE)

   apply merge_temp_box

   apply (erule STL4Edup)

    apply assumption

   apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])

   apply (rule classical)

   apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")

    apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])

   apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])

   apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])

   apply merge_act_box

   apply (frule 4 [temp_use])

      apply assumption+

   apply (erule_tac V = "sigmaa |= []F" in thin_rl)

   apply (drule BoxSFI [temp_use])

   apply (erule_tac F = "TEMP <>Enabled (<M>_g)" in dup_boxE)

   apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)

   apply merge_temp_box

   apply (erule DmdImpldup)

    apply assumption

   apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]

     SF_Box [try_rewrite] box_stp_act [try_rewrite])

    apply (force elim!: TLA2E [where P = P, temp_use])

   apply (rule STL2 [temp_use])

   apply (force simp: SF_def split_box_conj [try_rewrite]

     elim!: mp InfImpl [temp_use] intro!: 3 [temp_use])

   done

 (* ------------------------------------------------------------------------- *)

 (***           Liveness proofs by well-founded orderings                   ***)

 (* ------------------------------------------------------------------------- *)

 section "Well-founded orderings"

 lemma wf_leadsto:

   assumes 1: "wf r"

     and 2: "!!x. sigma |= F x ~> (G | (EX y. #((y,x):r) & F y))    "

   shows "sigma |= F x ~> G"

   apply (rule 1 [THEN wf_induct])

   apply (rule LatticeTriangle [temp_use])

    apply (rule 2)

   apply (auto simp: leadsto_exists [try_rewrite])

   apply (case_tac "(y,x) :r")

    apply force

   apply (force simp: leadsto_def Init_simps intro!: necT [temp_use])

   done

 (* If r is well-founded, state function v cannot decrease forever *)

 lemma wf_not_box_decrease: "!!r. wf r ==> |- [][ (v`, $v) : #r ]_v --> <>[][#False]_v"

   apply clarsimp

   apply (rule ccontr)

   apply (subgoal_tac "sigma |= (EX x. v=#x) ~> #False")

    apply (drule leadsto_false [temp_use, THEN iffD1, THEN STL2_gen [temp_use]])

    apply (force simp: Init_defs)

   apply (clarsimp simp: leadsto_exists [try_rewrite] not_square [try_rewrite] more_temp_simps)

   apply (erule wf_leadsto)

   apply (rule ensures_simple [temp_use])

    apply (auto simp: square_def angle_def)

   done

 (* "wf r  ==>  |- <>[][ (v`, $v) : #r ]_v --> <>[][#False]_v" *)

 lemmas wf_not_dmd_box_decrease =

   wf_not_box_decrease [THEN DmdImpl, unfolded more_temp_simps]

 (* If there are infinitely many steps where v decreases, then there

    have to be infinitely many non-stuttering steps where v doesn't decrease.

 *)

 lemma wf_box_dmd_decrease:

   assumes 1: "wf r"

   shows "|- []<>((v`, $v) : #r) --> []<><(v`, $v) ~: #r>_v"

   apply clarsimp

   apply (rule ccontr)

   apply (simp add: not_angle [try_rewrite] more_temp_simps)

   apply (drule 1 [THEN wf_not_dmd_box_decrease [temp_use]])

   apply (drule BoxDmdDmdBox [temp_use])

    apply assumption

   apply (subgoal_tac "sigma |= []<> ((#False) ::action)")

    apply force

   apply (erule STL4E)

   apply (rule DmdImpl)

   apply (force intro: 1 [THEN wf_irrefl, temp_use])

   done

 (* In particular, for natural numbers, if n decreases infinitely often

    then it has to increase infinitely often.

 *)

 lemma nat_box_dmd_decrease: "!!n::nat stfun. |- []<>(n` < $n) --> []<>($n < n`)"

   apply clarsimp

   apply (subgoal_tac "sigma |= []<><~ ((n`,$n) : #less_than) >_n")

    apply (erule thin_rl)

    apply (erule STL4E)

    apply (rule DmdImpl)

    apply (clarsimp simp: angle_def [try_rewrite])

   apply (rule wf_box_dmd_decrease [temp_use])

    apply (auto elim!: STL4E [temp_use] DmdImplE [temp_use])

   done

 (* ------------------------------------------------------------------------- *)

 (***           Flexible quantification over state variables                ***)

 (* ------------------------------------------------------------------------- *)

 section "Flexible quantification"

 lemma aallI:

   assumes 1: "basevars vs"

     and 2: "(!!x. basevars (x,vs) ==> sigma |= F x)"

   shows "sigma |= (AALL x. F x)"

   by (auto simp: aall_def elim!: eexE [temp_use] intro!: 1 dest!: 2 [temp_use])

 lemma aallE: "|- (AALL x. F x) --> F x"

   apply (unfold aall_def)

   apply clarsimp

   apply (erule contrapos_np)

   apply (force intro!: eexI [temp_use])

   done

 (* monotonicity of quantification *)

 lemma eex_mono:

   assumes 1: "sigma |= EEX x. F x"

     and 2: "!!x. sigma |= F x --> G x"

   shows "sigma |= EEX x. G x"

   apply (rule unit_base [THEN 1 [THEN eexE]])

   apply (rule eexI [temp_use])

   apply (erule 2 [unfolded intensional_rews, THEN mp])

   done

 lemma aall_mono:

   assumes 1: "sigma |= AALL x. F(x)"

     and 2: "!!x. sigma |= F(x) --> G(x)"

   shows "sigma |= AALL x. G(x)"

   apply (rule unit_base [THEN aallI])

   apply (rule 2 [unfolded intensional_rews, THEN mp])

   apply (rule 1 [THEN aallE [temp_use]])

   done

 (* Derived history introduction rule *)

 lemma historyI:

   assumes 1: "sigma |= Init I"

     and 2: "sigma |= []N"

     and 3: "basevars vs"

     and 4: "!!h. basevars(h,vs) ==> |- I & h = ha --> HI h"

     and 5: "!!h s t. [| basevars(h,vs); N (s,t); h t = hb (h s) (s,t) |] ==> HN h (s,t)"

   shows "sigma |= EEX h. Init (HI h) & [](HN h)"

   apply (rule history [temp_use, THEN eexE])

   apply (rule 3)

   apply (rule eexI [temp_use])

   apply clarsimp

   apply (rule conjI)

    prefer 2

    apply (insert 2)

    apply merge_box

    apply (force elim!: STL4E [temp_use] 5 [temp_use])

   apply (insert 1)

   apply (force simp: Init_defs elim!: 4 [temp_use])

   done

 (* ----------------------------------------------------------------------

    example of a history variable: existence of a clock

 *)

 lemma "|- EEX h. Init(h = #True) & [](h` = (~$h))"

   apply (rule tempI)

   apply (rule historyI)

   apply (force simp: Init_defs intro!: unit_base [temp_use] necT [temp_use])+

   done

 end