src/HOL/TLA/TLA.thy
author wenzelm
Thu Apr 18 17:07:01 2013 +0200 (2013-04-18)
changeset 51717 9e7d1c139569
parent 51668 5e1108291c7f
child 54742 7a86358a3c0b
permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
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(*  Title:      HOL/TLA/TLA.thy
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    Author:     Stephan Merz
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    Copyright:  1998 University of Munich
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*)
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header {* The temporal level of TLA *}
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theory TLA
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imports Init
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begin
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consts
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  (** abstract syntax **)
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  Box        :: "('w::world) form => temporal"
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  Dmd        :: "('w::world) form => temporal"
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  leadsto    :: "['w::world form, 'v::world form] => temporal"
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  Stable     :: "stpred => temporal"
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  WF         :: "[action, 'a stfun] => temporal"
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  SF         :: "[action, 'a stfun] => temporal"
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  (* Quantification over (flexible) state variables *)
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  EEx        :: "('a stfun => temporal) => temporal"       (binder "Eex " 10)
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  AAll       :: "('a stfun => temporal) => temporal"       (binder "Aall " 10)
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  (** concrete syntax **)
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syntax
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  "_Box"     :: "lift => lift"                        ("([]_)" [40] 40)
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  "_Dmd"     :: "lift => lift"                        ("(<>_)" [40] 40)
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  "_leadsto" :: "[lift,lift] => lift"                 ("(_ ~> _)" [23,22] 22)
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  "_stable"  :: "lift => lift"                        ("(stable/ _)")
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  "_WF"      :: "[lift,lift] => lift"                 ("(WF'(_')'_(_))" [0,60] 55)
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  "_SF"      :: "[lift,lift] => lift"                 ("(SF'(_')'_(_))" [0,60] 55)
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  "_EEx"     :: "[idts, lift] => lift"                ("(3EEX _./ _)" [0,10] 10)
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  "_AAll"    :: "[idts, lift] => lift"                ("(3AALL _./ _)" [0,10] 10)
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translations
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  "_Box"      ==   "CONST Box"
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  "_Dmd"      ==   "CONST Dmd"
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  "_leadsto"  ==   "CONST leadsto"
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  "_stable"   ==   "CONST Stable"
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  "_WF"       ==   "CONST WF"
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  "_SF"       ==   "CONST SF"
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  "_EEx v A"  ==   "Eex v. A"
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  "_AAll v A" ==   "Aall v. A"
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  "sigma |= []F"         <= "_Box F sigma"
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  "sigma |= <>F"         <= "_Dmd F sigma"
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  "sigma |= F ~> G"      <= "_leadsto F G sigma"
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  "sigma |= stable P"    <= "_stable P sigma"
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  "sigma |= WF(A)_v"     <= "_WF A v sigma"
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  "sigma |= SF(A)_v"     <= "_SF A v sigma"
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  "sigma |= EEX x. F"    <= "_EEx x F sigma"
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  "sigma |= AALL x. F"    <= "_AAll x F sigma"
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syntax (xsymbols)
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  "_Box"     :: "lift => lift"                        ("(\<box>_)" [40] 40)
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  "_Dmd"     :: "lift => lift"                        ("(\<diamond>_)" [40] 40)
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  "_leadsto" :: "[lift,lift] => lift"                 ("(_ \<leadsto> _)" [23,22] 22)
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  "_EEx"     :: "[idts, lift] => lift"                ("(3\<exists>\<exists> _./ _)" [0,10] 10)
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  "_AAll"    :: "[idts, lift] => lift"                ("(3\<forall>\<forall> _./ _)" [0,10] 10)
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syntax (HTML output)
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  "_EEx"     :: "[idts, lift] => lift"                ("(3\<exists>\<exists> _./ _)" [0,10] 10)
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  "_AAll"    :: "[idts, lift] => lift"                ("(3\<forall>\<forall> _./ _)" [0,10] 10)
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axiomatization where
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  (* Definitions of derived operators *)
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  dmd_def:      "\<And>F. TEMP <>F  ==  TEMP ~[]~F"
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axiomatization where
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  boxInit:      "\<And>F. TEMP []F  ==  TEMP []Init F" and
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  leadsto_def:  "\<And>F G. TEMP F ~> G  ==  TEMP [](Init F --> <>G)" and
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  stable_def:   "\<And>P. TEMP stable P  ==  TEMP []($P --> P$)" and
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  WF_def:       "TEMP WF(A)_v  ==  TEMP <>[] Enabled(<A>_v) --> []<><A>_v" and
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  SF_def:       "TEMP SF(A)_v  ==  TEMP []<> Enabled(<A>_v) --> []<><A>_v" and
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  aall_def:     "TEMP (AALL x. F x)  ==  TEMP ~ (EEX x. ~ F x)"
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axiomatization where
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(* Base axioms for raw TLA. *)
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  normalT:    "\<And>F G. |- [](F --> G) --> ([]F --> []G)" and    (* polymorphic *)
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  reflT:      "\<And>F. |- []F --> F" and         (* F::temporal *)
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  transT:     "\<And>F. |- []F --> [][]F" and     (* polymorphic *)
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  linT:       "\<And>F G. |- <>F & <>G --> (<>(F & <>G)) | (<>(G & <>F))" and
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  discT:      "\<And>F. |- [](F --> <>(~F & <>F)) --> (F --> []<>F)" and
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  primeI:     "\<And>P. |- []P --> Init P`" and
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  primeE:     "\<And>P F. |- [](Init P --> []F) --> Init P` --> (F --> []F)" and
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  indT:       "\<And>P F. |- [](Init P & ~[]F --> Init P` & F) --> Init P --> []F" and
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  allT:       "\<And>F. |- (ALL x. [](F x)) = ([](ALL x. F x))"
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axiomatization where
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  necT:       "\<And>F. |- F ==> |- []F"      (* polymorphic *)
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axiomatization where
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(* Flexible quantification: refinement mappings, history variables *)
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  eexI:       "|- F x --> (EEX x. F x)" and
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  eexE:       "[| sigma |= (EEX x. F x); basevars vs;
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                 (!!x. [| basevars (x, vs); sigma |= F x |] ==> (G sigma)::bool)
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              |] ==> G sigma" and
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  history:    "|- EEX h. Init(h = ha) & [](!x. $h = #x --> h` = hb x)"
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(* Specialize intensional introduction/elimination rules for temporal formulas *)
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lemma tempI [intro!]: "(!!sigma. sigma |= (F::temporal)) ==> |- F"
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  apply (rule intI)
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  apply (erule meta_spec)
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  done
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lemma tempD [dest]: "|- (F::temporal) ==> sigma |= F"
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  by (erule intD)
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(* ======== Functions to "unlift" temporal theorems ====== *)
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ML {*
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(* The following functions are specialized versions of the corresponding
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   functions defined in theory Intensional in that they introduce a
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   "world" parameter of type "behavior".
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*)
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fun temp_unlift th =
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  (rewrite_rule @{thms action_rews} (th RS @{thm tempD})) handle THM _ => action_unlift th;
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(* Turn  |- F = G  into meta-level rewrite rule  F == G *)
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val temp_rewrite = int_rewrite
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fun temp_use th =
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  case (concl_of th) of
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    Const _ $ (Const (@{const_name Intensional.Valid}, _) $ _) =>
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            ((flatten (temp_unlift th)) handle THM _ => th)
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  | _ => th;
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fun try_rewrite th = temp_rewrite th handle THM _ => temp_use th;
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*}
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attribute_setup temp_unlift = {* Scan.succeed (Thm.rule_attribute (K temp_unlift)) *}
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attribute_setup temp_rewrite = {* Scan.succeed (Thm.rule_attribute (K temp_rewrite)) *}
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attribute_setup temp_use = {* Scan.succeed (Thm.rule_attribute (K temp_use)) *}
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attribute_setup try_rewrite = {* Scan.succeed (Thm.rule_attribute (K try_rewrite)) *}
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(* ------------------------------------------------------------------------- *)
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(***           "Simple temporal logic": only [] and <>                     ***)
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(* ------------------------------------------------------------------------- *)
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section "Simple temporal logic"
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(* []~F == []~Init F *)
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lemmas boxNotInit = boxInit [of "LIFT ~F", unfolded Init_simps] for F
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lemma dmdInit: "TEMP <>F == TEMP <> Init F"
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  apply (unfold dmd_def)
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  apply (unfold boxInit [of "LIFT ~F"])
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  apply (simp (no_asm) add: Init_simps)
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  done
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lemmas dmdNotInit = dmdInit [of "LIFT ~F", unfolded Init_simps] for F
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(* boxInit and dmdInit cannot be used as rewrites, because they loop.
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   Non-looping instances for state predicates and actions are occasionally useful.
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*)
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lemmas boxInit_stp = boxInit [where 'a = state]
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lemmas boxInit_act = boxInit [where 'a = "state * state"]
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lemmas dmdInit_stp = dmdInit [where 'a = state]
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lemmas dmdInit_act = dmdInit [where 'a = "state * state"]
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(* The symmetric equations can be used to get rid of Init *)
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lemmas boxInitD = boxInit [symmetric]
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lemmas dmdInitD = dmdInit [symmetric]
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lemmas boxNotInitD = boxNotInit [symmetric]
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lemmas dmdNotInitD = dmdNotInit [symmetric]
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lemmas Init_simps = Init_simps boxInitD dmdInitD boxNotInitD dmdNotInitD
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(* ------------------------ STL2 ------------------------------------------- *)
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lemmas STL2 = reflT
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(* The "polymorphic" (generic) variant *)
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lemma STL2_gen: "|- []F --> Init F"
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  apply (unfold boxInit [of F])
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  apply (rule STL2)
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  done
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(* see also STL2_pr below: "|- []P --> Init P & Init (P`)" *)
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(* Dual versions for <> *)
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lemma InitDmd: "|- F --> <> F"
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  apply (unfold dmd_def)
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  apply (auto dest!: STL2 [temp_use])
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  done
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lemma InitDmd_gen: "|- Init F --> <>F"
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  apply clarsimp
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  apply (drule InitDmd [temp_use])
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  apply (simp add: dmdInitD)
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  done
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(* ------------------------ STL3 ------------------------------------------- *)
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lemma STL3: "|- ([][]F) = ([]F)"
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  by (auto elim: transT [temp_use] STL2 [temp_use])
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(* corresponding elimination rule introduces double boxes:
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   [| (sigma |= []F); (sigma |= [][]F) ==> PROP W |] ==> PROP W
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*)
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lemmas dup_boxE = STL3 [temp_unlift, THEN iffD2, elim_format]
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lemmas dup_boxD = STL3 [temp_unlift, THEN iffD1]
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(* dual versions for <> *)
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lemma DmdDmd: "|- (<><>F) = (<>F)"
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  by (auto simp add: dmd_def [try_rewrite] STL3 [try_rewrite])
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lemmas dup_dmdE = DmdDmd [temp_unlift, THEN iffD2, elim_format]
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lemmas dup_dmdD = DmdDmd [temp_unlift, THEN iffD1]
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(* ------------------------ STL4 ------------------------------------------- *)
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lemma STL4:
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  assumes "|- F --> G"
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  shows "|- []F --> []G"
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  apply clarsimp
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  apply (rule normalT [temp_use])
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   apply (rule assms [THEN necT, temp_use])
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  apply assumption
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  done
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(* Unlifted version as an elimination rule *)
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lemma STL4E: "[| sigma |= []F; |- F --> G |] ==> sigma |= []G"
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  by (erule (1) STL4 [temp_use])
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lemma STL4_gen: "|- Init F --> Init G ==> |- []F --> []G"
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  apply (drule STL4)
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  apply (simp add: boxInitD)
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  done
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lemma STL4E_gen: "[| sigma |= []F; |- Init F --> Init G |] ==> sigma |= []G"
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  by (erule (1) STL4_gen [temp_use])
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(* see also STL4Edup below, which allows an auxiliary boxed formula:
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       []A /\ F => G
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     -----------------
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     []A /\ []F => []G
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*)
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(* The dual versions for <> *)
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lemma DmdImpl:
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  assumes prem: "|- F --> G"
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  shows "|- <>F --> <>G"
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  apply (unfold dmd_def)
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  apply (fastforce intro!: prem [temp_use] elim!: STL4E [temp_use])
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  done
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lemma DmdImplE: "[| sigma |= <>F; |- F --> G |] ==> sigma |= <>G"
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  by (erule (1) DmdImpl [temp_use])
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(* ------------------------ STL5 ------------------------------------------- *)
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lemma STL5: "|- ([]F & []G) = ([](F & G))"
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  apply auto
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  apply (subgoal_tac "sigma |= [] (G --> (F & G))")
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     apply (erule normalT [temp_use])
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     apply (fastforce elim!: STL4E [temp_use])+
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  done
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(* rewrite rule to split conjunctions under boxes *)
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lemmas split_box_conj = STL5 [temp_unlift, symmetric]
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(* the corresponding elimination rule allows to combine boxes in the hypotheses
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   (NB: F and G must have the same type, i.e., both actions or temporals.)
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   Use "addSE2" etc. if you want to add this to a claset, otherwise it will loop!
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*)
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lemma box_conjE:
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  assumes "sigma |= []F"
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     and "sigma |= []G"
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  and "sigma |= [](F&G) ==> PROP R"
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  shows "PROP R"
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  by (rule assms STL5 [temp_unlift, THEN iffD1] conjI)+
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(* Instances of box_conjE for state predicates, actions, and temporals
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   in case the general rule is "too polymorphic".
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*)
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lemmas box_conjE_temp = box_conjE [where 'a = behavior]
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lemmas box_conjE_stp = box_conjE [where 'a = state]
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lemmas box_conjE_act = box_conjE [where 'a = "state * state"]
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(* Define a tactic that tries to merge all boxes in an antecedent. The definition is
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   a bit kludgy in order to simulate "double elim-resolution".
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*)
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lemma box_thin: "[| sigma |= []F; PROP W |] ==> PROP W" .
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ML {*
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fun merge_box_tac i =
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   REPEAT_DETERM (EVERY [etac @{thm box_conjE} i, atac i, etac @{thm box_thin} i])
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fun merge_temp_box_tac ctxt i =
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   REPEAT_DETERM (EVERY [etac @{thm box_conjE_temp} i, atac i,
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                         eres_inst_tac ctxt [(("'a", 0), "behavior")] @{thm box_thin} i])
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fun merge_stp_box_tac ctxt i =
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   REPEAT_DETERM (EVERY [etac @{thm box_conjE_stp} i, atac i,
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                         eres_inst_tac ctxt [(("'a", 0), "state")] @{thm box_thin} i])
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fun merge_act_box_tac ctxt i =
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   REPEAT_DETERM (EVERY [etac @{thm box_conjE_act} i, atac i,
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                         eres_inst_tac ctxt [(("'a", 0), "state * state")] @{thm box_thin} i])
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*}
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method_setup merge_box = {* Scan.succeed (K (SIMPLE_METHOD' merge_box_tac)) *}
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method_setup merge_temp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_temp_box_tac) *}
wenzelm@42814
   311
method_setup merge_stp_box = {* Scan.succeed (SIMPLE_METHOD' o merge_stp_box_tac) *}
wenzelm@42814
   312
method_setup merge_act_box = {* Scan.succeed (SIMPLE_METHOD' o merge_act_box_tac) *}
wenzelm@42787
   313
wenzelm@21624
   314
(* rewrite rule to push universal quantification through box:
wenzelm@21624
   315
      (sigma |= [](! x. F x)) = (! x. (sigma |= []F x))
wenzelm@21624
   316
*)
wenzelm@45605
   317
lemmas all_box = allT [temp_unlift, symmetric]
wenzelm@21624
   318
wenzelm@21624
   319
lemma DmdOr: "|- (<>(F | G)) = (<>F | <>G)"
wenzelm@21624
   320
  apply (auto simp add: dmd_def split_box_conj [try_rewrite])
nipkow@44890
   321
  apply (erule contrapos_np, merge_box, fastforce elim!: STL4E [temp_use])+
wenzelm@21624
   322
  done
wenzelm@21624
   323
wenzelm@21624
   324
lemma exT: "|- (EX x. <>(F x)) = (<>(EX x. F x))"
wenzelm@21624
   325
  by (auto simp: dmd_def Not_Rex [try_rewrite] all_box [try_rewrite])
wenzelm@21624
   326
wenzelm@45605
   327
lemmas ex_dmd = exT [temp_unlift, symmetric]
wenzelm@21624
   328
wenzelm@21624
   329
lemma STL4Edup: "!!sigma. [| sigma |= []A; sigma |= []F; |- F & []A --> G |] ==> sigma |= []G"
wenzelm@21624
   330
  apply (erule dup_boxE)
wenzelm@42787
   331
  apply merge_box
wenzelm@21624
   332
  apply (erule STL4E)
wenzelm@21624
   333
  apply assumption
wenzelm@21624
   334
  done
wenzelm@21624
   335
wenzelm@21624
   336
lemma DmdImpl2: 
wenzelm@21624
   337
    "!!sigma. [| sigma |= <>F; sigma |= [](F --> G) |] ==> sigma |= <>G"
wenzelm@21624
   338
  apply (unfold dmd_def)
wenzelm@21624
   339
  apply auto
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   340
  apply (erule notE)
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   341
  apply merge_box
nipkow@44890
   342
  apply (fastforce elim!: STL4E [temp_use])
wenzelm@21624
   343
  done
wenzelm@21624
   344
wenzelm@21624
   345
lemma InfImpl:
wenzelm@21624
   346
  assumes 1: "sigma |= []<>F"
wenzelm@21624
   347
    and 2: "sigma |= []G"
wenzelm@21624
   348
    and 3: "|- F & G --> H"
wenzelm@21624
   349
  shows "sigma |= []<>H"
wenzelm@21624
   350
  apply (insert 1 2)
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   351
  apply (erule_tac F = G in dup_boxE)
wenzelm@42787
   352
  apply merge_box
nipkow@44890
   353
  apply (fastforce elim!: STL4E [temp_use] DmdImpl2 [temp_use] intro!: 3 [temp_use])
wenzelm@21624
   354
  done
wenzelm@21624
   355
wenzelm@21624
   356
(* ------------------------ STL6 ------------------------------------------- *)
wenzelm@21624
   357
(* Used in the proof of STL6, but useful in itself. *)
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   358
lemma BoxDmd: "|- []F & <>G --> <>([]F & G)"
wenzelm@21624
   359
  apply (unfold dmd_def)
wenzelm@21624
   360
  apply clarsimp
wenzelm@21624
   361
  apply (erule dup_boxE)
wenzelm@42787
   362
  apply merge_box
wenzelm@21624
   363
  apply (erule contrapos_np)
nipkow@44890
   364
  apply (fastforce elim!: STL4E [temp_use])
wenzelm@21624
   365
  done
wenzelm@21624
   366
wenzelm@21624
   367
(* weaker than BoxDmd, but more polymorphic (and often just right) *)
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   368
lemma BoxDmd_simple: "|- []F & <>G --> <>(F & G)"
wenzelm@21624
   369
  apply (unfold dmd_def)
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   370
  apply clarsimp
wenzelm@42787
   371
  apply merge_box
nipkow@44890
   372
  apply (fastforce elim!: notE STL4E [temp_use])
wenzelm@21624
   373
  done
wenzelm@21624
   374
wenzelm@21624
   375
lemma BoxDmd2_simple: "|- []F & <>G --> <>(G & F)"
wenzelm@21624
   376
  apply (unfold dmd_def)
wenzelm@21624
   377
  apply clarsimp
wenzelm@42787
   378
  apply merge_box
nipkow@44890
   379
  apply (fastforce elim!: notE STL4E [temp_use])
wenzelm@21624
   380
  done
wenzelm@21624
   381
wenzelm@21624
   382
lemma DmdImpldup:
wenzelm@21624
   383
  assumes 1: "sigma |= []A"
wenzelm@21624
   384
    and 2: "sigma |= <>F"
wenzelm@21624
   385
    and 3: "|- []A & F --> G"
wenzelm@21624
   386
  shows "sigma |= <>G"
wenzelm@21624
   387
  apply (rule 2 [THEN 1 [THEN BoxDmd [temp_use]], THEN DmdImplE])
wenzelm@21624
   388
  apply (rule 3)
wenzelm@21624
   389
  done
wenzelm@21624
   390
wenzelm@21624
   391
lemma STL6: "|- <>[]F & <>[]G --> <>[](F & G)"
wenzelm@21624
   392
  apply (auto simp: STL5 [temp_rewrite, symmetric])
wenzelm@21624
   393
  apply (drule linT [temp_use])
wenzelm@21624
   394
   apply assumption
wenzelm@21624
   395
  apply (erule thin_rl)
wenzelm@21624
   396
  apply (rule DmdDmd [temp_unlift, THEN iffD1])
wenzelm@21624
   397
  apply (erule disjE)
wenzelm@21624
   398
   apply (erule DmdImplE)
wenzelm@21624
   399
   apply (rule BoxDmd)
wenzelm@21624
   400
  apply (erule DmdImplE)
wenzelm@21624
   401
  apply auto
wenzelm@21624
   402
  apply (drule BoxDmd [temp_use])
wenzelm@21624
   403
   apply assumption
wenzelm@21624
   404
  apply (erule thin_rl)
nipkow@44890
   405
  apply (fastforce elim!: DmdImplE [temp_use])
wenzelm@21624
   406
  done
wenzelm@21624
   407
wenzelm@21624
   408
wenzelm@21624
   409
(* ------------------------ True / False ----------------------------------------- *)
wenzelm@21624
   410
section "Simplification of constants"
wenzelm@21624
   411
wenzelm@21624
   412
lemma BoxConst: "|- ([]#P) = #P"
wenzelm@21624
   413
  apply (rule tempI)
wenzelm@21624
   414
  apply (cases P)
wenzelm@21624
   415
   apply (auto intro!: necT [temp_use] dest: STL2_gen [temp_use] simp: Init_simps)
wenzelm@21624
   416
  done
wenzelm@21624
   417
wenzelm@21624
   418
lemma DmdConst: "|- (<>#P) = #P"
wenzelm@21624
   419
  apply (unfold dmd_def)
wenzelm@21624
   420
  apply (cases P)
wenzelm@21624
   421
  apply (simp_all add: BoxConst [try_rewrite])
wenzelm@21624
   422
  done
wenzelm@21624
   423
wenzelm@21624
   424
lemmas temp_simps [temp_rewrite, simp] = BoxConst DmdConst
wenzelm@21624
   425
wenzelm@21624
   426
wenzelm@21624
   427
(* ------------------------ Further rewrites ----------------------------------------- *)
wenzelm@21624
   428
section "Further rewrites"
wenzelm@21624
   429
wenzelm@21624
   430
lemma NotBox: "|- (~[]F) = (<>~F)"
wenzelm@21624
   431
  by (simp add: dmd_def)
wenzelm@21624
   432
wenzelm@21624
   433
lemma NotDmd: "|- (~<>F) = ([]~F)"
wenzelm@21624
   434
  by (simp add: dmd_def)
wenzelm@21624
   435
wenzelm@21624
   436
(* These are not declared by default, because they could be harmful,
wenzelm@21624
   437
   e.g. []F & ~[]F becomes []F & <>~F !! *)
wenzelm@26305
   438
lemmas more_temp_simps1 =
wenzelm@21624
   439
  STL3 [temp_rewrite] DmdDmd [temp_rewrite] NotBox [temp_rewrite] NotDmd [temp_rewrite]
wenzelm@21624
   440
  NotBox [temp_unlift, THEN eq_reflection]
wenzelm@21624
   441
  NotDmd [temp_unlift, THEN eq_reflection]
wenzelm@21624
   442
wenzelm@21624
   443
lemma BoxDmdBox: "|- ([]<>[]F) = (<>[]F)"
wenzelm@21624
   444
  apply (auto dest!: STL2 [temp_use])
wenzelm@21624
   445
  apply (rule ccontr)
wenzelm@21624
   446
  apply (subgoal_tac "sigma |= <>[][]F & <>[]~[]F")
wenzelm@21624
   447
   apply (erule thin_rl)
wenzelm@21624
   448
   apply auto
wenzelm@21624
   449
    apply (drule STL6 [temp_use])
wenzelm@21624
   450
     apply assumption
wenzelm@21624
   451
    apply simp
wenzelm@26305
   452
   apply (simp_all add: more_temp_simps1)
wenzelm@21624
   453
  done
wenzelm@21624
   454
wenzelm@21624
   455
lemma DmdBoxDmd: "|- (<>[]<>F) = ([]<>F)"
wenzelm@21624
   456
  apply (unfold dmd_def)
wenzelm@21624
   457
  apply (auto simp: BoxDmdBox [unfolded dmd_def, try_rewrite])
wenzelm@21624
   458
  done
wenzelm@21624
   459
wenzelm@26305
   460
lemmas more_temp_simps2 = more_temp_simps1 BoxDmdBox [temp_rewrite] DmdBoxDmd [temp_rewrite]
wenzelm@21624
   461
wenzelm@21624
   462
wenzelm@21624
   463
(* ------------------------ Miscellaneous ----------------------------------- *)
wenzelm@21624
   464
wenzelm@21624
   465
lemma BoxOr: "!!sigma. [| sigma |= []F | []G |] ==> sigma |= [](F | G)"
nipkow@44890
   466
  by (fastforce elim!: STL4E [temp_use])
wenzelm@21624
   467
wenzelm@21624
   468
(* "persistently implies infinitely often" *)
wenzelm@21624
   469
lemma DBImplBD: "|- <>[]F --> []<>F"
wenzelm@21624
   470
  apply clarsimp
wenzelm@21624
   471
  apply (rule ccontr)
wenzelm@26305
   472
  apply (simp add: more_temp_simps2)
wenzelm@21624
   473
  apply (drule STL6 [temp_use])
wenzelm@21624
   474
   apply assumption
wenzelm@21624
   475
  apply simp
wenzelm@21624
   476
  done
wenzelm@21624
   477
wenzelm@21624
   478
lemma BoxDmdDmdBox: "|- []<>F & <>[]G --> []<>(F & G)"
wenzelm@21624
   479
  apply clarsimp
wenzelm@21624
   480
  apply (rule ccontr)
wenzelm@26305
   481
  apply (unfold more_temp_simps2)
wenzelm@21624
   482
  apply (drule STL6 [temp_use])
wenzelm@21624
   483
   apply assumption
wenzelm@21624
   484
  apply (subgoal_tac "sigma |= <>[]~F")
wenzelm@21624
   485
   apply (force simp: dmd_def)
nipkow@44890
   486
  apply (fastforce elim: DmdImplE [temp_use] STL4E [temp_use])
wenzelm@21624
   487
  done
wenzelm@21624
   488
wenzelm@21624
   489
wenzelm@21624
   490
(* ------------------------------------------------------------------------- *)
wenzelm@21624
   491
(***          TLA-specific theorems: primed formulas                       ***)
wenzelm@21624
   492
(* ------------------------------------------------------------------------- *)
wenzelm@21624
   493
section "priming"
wenzelm@21624
   494
wenzelm@21624
   495
(* ------------------------ TLA2 ------------------------------------------- *)
wenzelm@21624
   496
lemma STL2_pr: "|- []P --> Init P & Init P`"
nipkow@44890
   497
  by (fastforce intro!: STL2_gen [temp_use] primeI [temp_use])
wenzelm@21624
   498
wenzelm@21624
   499
(* Auxiliary lemma allows priming of boxed actions *)
wenzelm@21624
   500
lemma BoxPrime: "|- []P --> []($P & P$)"
wenzelm@21624
   501
  apply clarsimp
wenzelm@21624
   502
  apply (erule dup_boxE)
wenzelm@21624
   503
  apply (unfold boxInit_act)
wenzelm@21624
   504
  apply (erule STL4E)
wenzelm@21624
   505
  apply (auto simp: Init_simps dest!: STL2_pr [temp_use])
wenzelm@21624
   506
  done
wenzelm@21624
   507
wenzelm@21624
   508
lemma TLA2:
wenzelm@21624
   509
  assumes "|- $P & P$ --> A"
wenzelm@21624
   510
  shows "|- []P --> []A"
wenzelm@21624
   511
  apply clarsimp
wenzelm@21624
   512
  apply (drule BoxPrime [temp_use])
wenzelm@41529
   513
  apply (auto simp: Init_stp_act_rev [try_rewrite] intro!: assms [temp_use]
wenzelm@21624
   514
    elim!: STL4E [temp_use])
wenzelm@21624
   515
  done
wenzelm@21624
   516
wenzelm@21624
   517
lemma TLA2E: "[| sigma |= []P; |- $P & P$ --> A |] ==> sigma |= []A"
wenzelm@21624
   518
  by (erule (1) TLA2 [temp_use])
wenzelm@21624
   519
wenzelm@21624
   520
lemma DmdPrime: "|- (<>P`) --> (<>P)"
wenzelm@21624
   521
  apply (unfold dmd_def)
nipkow@44890
   522
  apply (fastforce elim!: TLA2E [temp_use])
wenzelm@21624
   523
  done
wenzelm@21624
   524
wenzelm@45605
   525
lemmas PrimeDmd = InitDmd_gen [temp_use, THEN DmdPrime [temp_use]]
wenzelm@21624
   526
wenzelm@21624
   527
(* ------------------------ INV1, stable --------------------------------------- *)
wenzelm@21624
   528
section "stable, invariant"
wenzelm@21624
   529
wenzelm@21624
   530
lemma ind_rule:
wenzelm@21624
   531
   "[| sigma |= []H; sigma |= Init P; |- H --> (Init P & ~[]F --> Init(P`) & F) |]  
wenzelm@21624
   532
    ==> sigma |= []F"
wenzelm@21624
   533
  apply (rule indT [temp_use])
wenzelm@21624
   534
   apply (erule (2) STL4E)
wenzelm@21624
   535
  done
wenzelm@21624
   536
wenzelm@21624
   537
lemma box_stp_act: "|- ([]$P) = ([]P)"
wenzelm@21624
   538
  by (simp add: boxInit_act Init_simps)
wenzelm@21624
   539
wenzelm@45605
   540
lemmas box_stp_actI = box_stp_act [temp_use, THEN iffD2]
wenzelm@45605
   541
lemmas box_stp_actD = box_stp_act [temp_use, THEN iffD1]
wenzelm@21624
   542
wenzelm@26305
   543
lemmas more_temp_simps3 = box_stp_act [temp_rewrite] more_temp_simps2
wenzelm@21624
   544
wenzelm@21624
   545
lemma INV1: 
wenzelm@21624
   546
  "|- (Init P) --> (stable P) --> []P"
wenzelm@21624
   547
  apply (unfold stable_def boxInit_stp boxInit_act)
wenzelm@21624
   548
  apply clarsimp
wenzelm@21624
   549
  apply (erule ind_rule)
wenzelm@21624
   550
   apply (auto simp: Init_simps elim: ind_rule)
wenzelm@21624
   551
  done
wenzelm@21624
   552
wenzelm@21624
   553
lemma StableT: 
wenzelm@21624
   554
    "!!P. |- $P & A --> P` ==> |- []A --> stable P"
wenzelm@21624
   555
  apply (unfold stable_def)
nipkow@44890
   556
  apply (fastforce elim!: STL4E [temp_use])
wenzelm@21624
   557
  done
wenzelm@21624
   558
wenzelm@21624
   559
lemma Stable: "[| sigma |= []A; |- $P & A --> P` |] ==> sigma |= stable P"
wenzelm@21624
   560
  by (erule (1) StableT [temp_use])
wenzelm@21624
   561
wenzelm@21624
   562
(* Generalization of INV1 *)
wenzelm@21624
   563
lemma StableBox: "|- (stable P) --> [](Init P --> []P)"
wenzelm@21624
   564
  apply (unfold stable_def)
wenzelm@21624
   565
  apply clarsimp
wenzelm@21624
   566
  apply (erule dup_boxE)
wenzelm@21624
   567
  apply (force simp: stable_def elim: STL4E [temp_use] INV1 [temp_use])
wenzelm@21624
   568
  done
wenzelm@21624
   569
wenzelm@21624
   570
lemma DmdStable: "|- (stable P) & <>P --> <>[]P"
wenzelm@21624
   571
  apply clarsimp
wenzelm@21624
   572
  apply (rule DmdImpl2)
wenzelm@21624
   573
   prefer 2
wenzelm@21624
   574
   apply (erule StableBox [temp_use])
wenzelm@21624
   575
  apply (simp add: dmdInitD)
wenzelm@21624
   576
  done
wenzelm@21624
   577
wenzelm@21624
   578
(* ---------------- (Semi-)automatic invariant tactics ---------------------- *)
wenzelm@21624
   579
wenzelm@21624
   580
ML {*
wenzelm@21624
   581
(* inv_tac reduces goals of the form ... ==> sigma |= []P *)
wenzelm@42793
   582
fun inv_tac ctxt =
wenzelm@42793
   583
  SELECT_GOAL
wenzelm@42793
   584
    (EVERY
wenzelm@42793
   585
     [auto_tac ctxt,
wenzelm@42793
   586
      TRY (merge_box_tac 1),
wenzelm@42793
   587
      rtac (temp_use @{thm INV1}) 1, (* fail if the goal is not a box *)
wenzelm@42793
   588
      TRYALL (etac @{thm Stable})]);
wenzelm@21624
   589
wenzelm@21624
   590
(* auto_inv_tac applies inv_tac and then tries to attack the subgoals
wenzelm@21624
   591
   in simple cases it may be able to handle goals like |- MyProg --> []Inv.
wenzelm@21624
   592
   In these simple cases the simplifier seems to be more useful than the
wenzelm@21624
   593
   auto-tactic, which applies too much propositional logic and simplifies
wenzelm@21624
   594
   too late.
wenzelm@21624
   595
*)
wenzelm@42803
   596
fun auto_inv_tac ctxt =
wenzelm@42793
   597
  SELECT_GOAL
wenzelm@42803
   598
    (inv_tac ctxt 1 THEN
wenzelm@42793
   599
      (TRYALL (action_simp_tac
wenzelm@51717
   600
        (ctxt addsimps [@{thm Init_stp}, @{thm Init_act}]) [] [@{thm squareE}])));
wenzelm@21624
   601
*}
wenzelm@21624
   602
wenzelm@42769
   603
method_setup invariant = {*
wenzelm@42793
   604
  Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o inv_tac))
wenzelm@42814
   605
*}
wenzelm@42769
   606
wenzelm@42769
   607
method_setup auto_invariant = {*
wenzelm@42803
   608
  Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o auto_inv_tac))
wenzelm@42814
   609
*}
wenzelm@42769
   610
wenzelm@21624
   611
lemma unless: "|- []($P --> P` | Q`) --> (stable P) | <>Q"
wenzelm@21624
   612
  apply (unfold dmd_def)
wenzelm@21624
   613
  apply (clarsimp dest!: BoxPrime [temp_use])
wenzelm@42787
   614
  apply merge_box
wenzelm@21624
   615
  apply (erule contrapos_np)
nipkow@44890
   616
  apply (fastforce elim!: Stable [temp_use])
wenzelm@21624
   617
  done
wenzelm@21624
   618
wenzelm@21624
   619
wenzelm@21624
   620
(* --------------------- Recursive expansions --------------------------------------- *)
wenzelm@21624
   621
section "recursive expansions"
wenzelm@21624
   622
wenzelm@21624
   623
(* Recursive expansions of [] and <> for state predicates *)
wenzelm@21624
   624
lemma BoxRec: "|- ([]P) = (Init P & []P`)"
wenzelm@21624
   625
  apply (auto intro!: STL2_gen [temp_use])
nipkow@44890
   626
   apply (fastforce elim!: TLA2E [temp_use])
wenzelm@21624
   627
  apply (auto simp: stable_def elim!: INV1 [temp_use] STL4E [temp_use])
wenzelm@21624
   628
  done
wenzelm@21624
   629
wenzelm@21624
   630
lemma DmdRec: "|- (<>P) = (Init P | <>P`)"
wenzelm@21624
   631
  apply (unfold dmd_def BoxRec [temp_rewrite])
wenzelm@21624
   632
  apply (auto simp: Init_simps)
wenzelm@21624
   633
  done
wenzelm@21624
   634
wenzelm@21624
   635
lemma DmdRec2: "!!sigma. [| sigma |= <>P; sigma |= []~P` |] ==> sigma |= Init P"
wenzelm@21624
   636
  apply (force simp: DmdRec [temp_rewrite] dmd_def)
wenzelm@21624
   637
  done
wenzelm@21624
   638
wenzelm@21624
   639
lemma InfinitePrime: "|- ([]<>P) = ([]<>P`)"
wenzelm@21624
   640
  apply auto
wenzelm@21624
   641
   apply (rule classical)
wenzelm@21624
   642
   apply (rule DBImplBD [temp_use])
wenzelm@21624
   643
   apply (subgoal_tac "sigma |= <>[]P")
nipkow@44890
   644
    apply (fastforce elim!: DmdImplE [temp_use] TLA2E [temp_use])
wenzelm@21624
   645
   apply (subgoal_tac "sigma |= <>[] (<>P & []~P`)")
wenzelm@21624
   646
    apply (force simp: boxInit_stp [temp_use]
wenzelm@21624
   647
      elim!: DmdImplE [temp_use] STL4E [temp_use] DmdRec2 [temp_use])
wenzelm@26305
   648
   apply (force intro!: STL6 [temp_use] simp: more_temp_simps3)
nipkow@44890
   649
  apply (fastforce intro: DmdPrime [temp_use] elim!: STL4E [temp_use])
wenzelm@21624
   650
  done
wenzelm@21624
   651
wenzelm@21624
   652
lemma InfiniteEnsures:
wenzelm@21624
   653
  "[| sigma |= []N; sigma |= []<>A; |- A & N --> P` |] ==> sigma |= []<>P"
wenzelm@21624
   654
  apply (unfold InfinitePrime [temp_rewrite])
wenzelm@21624
   655
  apply (rule InfImpl)
wenzelm@21624
   656
    apply assumption+
wenzelm@21624
   657
  done
wenzelm@21624
   658
wenzelm@21624
   659
(* ------------------------ fairness ------------------------------------------- *)
wenzelm@21624
   660
section "fairness"
wenzelm@21624
   661
wenzelm@21624
   662
(* alternative definitions of fairness *)
wenzelm@21624
   663
lemma WF_alt: "|- WF(A)_v = ([]<>~Enabled(<A>_v) | []<><A>_v)"
wenzelm@21624
   664
  apply (unfold WF_def dmd_def)
nipkow@44890
   665
  apply fastforce
wenzelm@21624
   666
  done
wenzelm@21624
   667
wenzelm@21624
   668
lemma SF_alt: "|- SF(A)_v = (<>[]~Enabled(<A>_v) | []<><A>_v)"
wenzelm@21624
   669
  apply (unfold SF_def dmd_def)
nipkow@44890
   670
  apply fastforce
wenzelm@21624
   671
  done
wenzelm@21624
   672
wenzelm@21624
   673
(* theorems to "box" fairness conditions *)
wenzelm@21624
   674
lemma BoxWFI: "|- WF(A)_v --> []WF(A)_v"
wenzelm@26305
   675
  by (auto simp: WF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
wenzelm@21624
   676
wenzelm@21624
   677
lemma WF_Box: "|- ([]WF(A)_v) = WF(A)_v"
nipkow@44890
   678
  by (fastforce intro!: BoxWFI [temp_use] dest!: STL2 [temp_use])
wenzelm@21624
   679
wenzelm@21624
   680
lemma BoxSFI: "|- SF(A)_v --> []SF(A)_v"
wenzelm@26305
   681
  by (auto simp: SF_alt [try_rewrite] more_temp_simps3 intro!: BoxOr [temp_use])
wenzelm@21624
   682
wenzelm@21624
   683
lemma SF_Box: "|- ([]SF(A)_v) = SF(A)_v"
nipkow@44890
   684
  by (fastforce intro!: BoxSFI [temp_use] dest!: STL2 [temp_use])
wenzelm@21624
   685
wenzelm@26305
   686
lemmas more_temp_simps = more_temp_simps3 WF_Box [temp_rewrite] SF_Box [temp_rewrite]
wenzelm@21624
   687
wenzelm@21624
   688
lemma SFImplWF: "|- SF(A)_v --> WF(A)_v"
wenzelm@21624
   689
  apply (unfold SF_def WF_def)
nipkow@44890
   690
  apply (fastforce dest!: DBImplBD [temp_use])
wenzelm@21624
   691
  done
wenzelm@21624
   692
wenzelm@21624
   693
(* A tactic that "boxes" all fairness conditions. Apply more_temp_simps to "unbox". *)
wenzelm@21624
   694
ML {*
wenzelm@26305
   695
val box_fair_tac = SELECT_GOAL (REPEAT (dresolve_tac [@{thm BoxWFI}, @{thm BoxSFI}] 1))
wenzelm@21624
   696
*}
wenzelm@21624
   697
wenzelm@21624
   698
wenzelm@21624
   699
(* ------------------------------ leads-to ------------------------------ *)
wenzelm@21624
   700
wenzelm@21624
   701
section "~>"
wenzelm@21624
   702
wenzelm@21624
   703
lemma leadsto_init: "|- (Init F) & (F ~> G) --> <>G"
wenzelm@21624
   704
  apply (unfold leadsto_def)
wenzelm@21624
   705
  apply (auto dest!: STL2 [temp_use])
wenzelm@21624
   706
  done
wenzelm@21624
   707
wenzelm@21624
   708
(* |- F & (F ~> G) --> <>G *)
wenzelm@45605
   709
lemmas leadsto_init_temp = leadsto_init [where 'a = behavior, unfolded Init_simps]
wenzelm@21624
   710
wenzelm@21624
   711
lemma streett_leadsto: "|- ([]<>Init F --> []<>G) = (<>(F ~> G))"
wenzelm@21624
   712
  apply (unfold leadsto_def)
wenzelm@21624
   713
  apply auto
wenzelm@21624
   714
    apply (simp add: more_temp_simps)
nipkow@44890
   715
    apply (fastforce elim!: DmdImplE [temp_use] STL4E [temp_use])
nipkow@44890
   716
   apply (fastforce intro!: InitDmd [temp_use] elim!: STL4E [temp_use])
wenzelm@21624
   717
  apply (subgoal_tac "sigma |= []<><>G")
wenzelm@21624
   718
   apply (simp add: more_temp_simps)
wenzelm@21624
   719
  apply (drule BoxDmdDmdBox [temp_use])
wenzelm@21624
   720
   apply assumption
nipkow@44890
   721
  apply (fastforce elim!: DmdImplE [temp_use] STL4E [temp_use])
wenzelm@21624
   722
  done
wenzelm@21624
   723
wenzelm@21624
   724
lemma leadsto_infinite: "|- []<>F & (F ~> G) --> []<>G"
wenzelm@21624
   725
  apply clarsimp
wenzelm@21624
   726
  apply (erule InitDmd [temp_use, THEN streett_leadsto [temp_unlift, THEN iffD2, THEN mp]])
wenzelm@21624
   727
  apply (simp add: dmdInitD)
wenzelm@21624
   728
  done
wenzelm@21624
   729
wenzelm@21624
   730
(* In particular, strong fairness is a Streett condition. The following
wenzelm@21624
   731
   rules are sometimes easier to use than WF2 or SF2 below.
wenzelm@21624
   732
*)
wenzelm@21624
   733
lemma leadsto_SF: "|- (Enabled(<A>_v) ~> <A>_v) --> SF(A)_v"
wenzelm@21624
   734
  apply (unfold SF_def)
wenzelm@21624
   735
  apply (clarsimp elim!: leadsto_infinite [temp_use])
wenzelm@21624
   736
  done
wenzelm@21624
   737
wenzelm@21624
   738
lemma leadsto_WF: "|- (Enabled(<A>_v) ~> <A>_v) --> WF(A)_v"
wenzelm@21624
   739
  by (clarsimp intro!: SFImplWF [temp_use] leadsto_SF [temp_use])
wenzelm@21624
   740
wenzelm@21624
   741
(* introduce an invariant into the proof of a leadsto assertion.
wenzelm@21624
   742
   []I --> ((P ~> Q)  =  (P /\ I ~> Q))
wenzelm@21624
   743
*)
wenzelm@21624
   744
lemma INV_leadsto: "|- []I & (P & I ~> Q) --> (P ~> Q)"
wenzelm@21624
   745
  apply (unfold leadsto_def)
wenzelm@21624
   746
  apply clarsimp
wenzelm@21624
   747
  apply (erule STL4Edup)
wenzelm@21624
   748
   apply assumption
wenzelm@21624
   749
  apply (auto simp: Init_simps dest!: STL2_gen [temp_use])
wenzelm@21624
   750
  done
wenzelm@21624
   751
wenzelm@21624
   752
lemma leadsto_classical: "|- (Init F & []~G ~> G) --> (F ~> G)"
wenzelm@21624
   753
  apply (unfold leadsto_def dmd_def)
wenzelm@21624
   754
  apply (force simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   755
  done
wenzelm@21624
   756
wenzelm@21624
   757
lemma leadsto_false: "|- (F ~> #False) = ([]~F)"
wenzelm@21624
   758
  apply (unfold leadsto_def)
wenzelm@21624
   759
  apply (simp add: boxNotInitD)
wenzelm@21624
   760
  done
wenzelm@21624
   761
wenzelm@21624
   762
lemma leadsto_exists: "|- ((EX x. F x) ~> G) = (ALL x. (F x ~> G))"
wenzelm@21624
   763
  apply (unfold leadsto_def)
wenzelm@21624
   764
  apply (auto simp: allT [try_rewrite] Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   765
  done
wenzelm@21624
   766
wenzelm@21624
   767
(* basic leadsto properties, cf. Unity *)
wenzelm@21624
   768
wenzelm@21624
   769
lemma ImplLeadsto_gen: "|- [](Init F --> Init G) --> (F ~> G)"
wenzelm@21624
   770
  apply (unfold leadsto_def)
wenzelm@21624
   771
  apply (auto intro!: InitDmd_gen [temp_use]
wenzelm@21624
   772
    elim!: STL4E_gen [temp_use] simp: Init_simps)
wenzelm@21624
   773
  done
wenzelm@21624
   774
wenzelm@45605
   775
lemmas ImplLeadsto =
wenzelm@45605
   776
  ImplLeadsto_gen [where 'a = behavior and 'b = behavior, unfolded Init_simps]
wenzelm@21624
   777
wenzelm@21624
   778
lemma ImplLeadsto_simple: "!!F G. |- F --> G ==> |- F ~> G"
wenzelm@21624
   779
  by (auto simp: Init_def intro!: ImplLeadsto_gen [temp_use] necT [temp_use])
wenzelm@21624
   780
wenzelm@21624
   781
lemma EnsuresLeadsto:
wenzelm@21624
   782
  assumes "|- A & $P --> Q`"
wenzelm@21624
   783
  shows "|- []A --> (P ~> Q)"
wenzelm@21624
   784
  apply (unfold leadsto_def)
wenzelm@21624
   785
  apply (clarsimp elim!: INV_leadsto [temp_use])
wenzelm@21624
   786
  apply (erule STL4E_gen)
wenzelm@21624
   787
  apply (auto simp: Init_defs intro!: PrimeDmd [temp_use] assms [temp_use])
wenzelm@21624
   788
  done
wenzelm@21624
   789
wenzelm@21624
   790
lemma EnsuresLeadsto2: "|- []($P --> Q`) --> (P ~> Q)"
wenzelm@21624
   791
  apply (unfold leadsto_def)
wenzelm@21624
   792
  apply clarsimp
wenzelm@21624
   793
  apply (erule STL4E_gen)
wenzelm@21624
   794
  apply (auto simp: Init_simps intro!: PrimeDmd [temp_use])
wenzelm@21624
   795
  done
wenzelm@21624
   796
wenzelm@21624
   797
lemma ensures:
wenzelm@21624
   798
  assumes 1: "|- $P & N --> P` | Q`"
wenzelm@21624
   799
    and 2: "|- ($P & N) & A --> Q`"
wenzelm@21624
   800
  shows "|- []N & []([]P --> <>A) --> (P ~> Q)"
wenzelm@21624
   801
  apply (unfold leadsto_def)
wenzelm@21624
   802
  apply clarsimp
wenzelm@21624
   803
  apply (erule STL4Edup)
wenzelm@21624
   804
   apply assumption
wenzelm@21624
   805
  apply clarsimp
wenzelm@21624
   806
  apply (subgoal_tac "sigmaa |= [] ($P --> P` | Q`) ")
wenzelm@21624
   807
   apply (drule unless [temp_use])
wenzelm@21624
   808
   apply (clarsimp dest!: INV1 [temp_use])
wenzelm@21624
   809
  apply (rule 2 [THEN DmdImpl, temp_use, THEN DmdPrime [temp_use]])
wenzelm@21624
   810
   apply (force intro!: BoxDmd_simple [temp_use]
wenzelm@21624
   811
     simp: split_box_conj [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
   812
  apply (force elim: STL4E [temp_use] dest: 1 [temp_use])
wenzelm@21624
   813
  done
wenzelm@21624
   814
wenzelm@21624
   815
lemma ensures_simple:
wenzelm@21624
   816
  "[| |- $P & N --> P` | Q`;  
wenzelm@21624
   817
      |- ($P & N) & A --> Q`  
wenzelm@21624
   818
   |] ==> |- []N & []<>A --> (P ~> Q)"
wenzelm@21624
   819
  apply clarsimp
wenzelm@21624
   820
  apply (erule (2) ensures [temp_use])
wenzelm@21624
   821
  apply (force elim!: STL4E [temp_use])
wenzelm@21624
   822
  done
wenzelm@21624
   823
wenzelm@21624
   824
lemma EnsuresInfinite:
wenzelm@21624
   825
    "[| sigma |= []<>P; sigma |= []A; |- A & $P --> Q` |] ==> sigma |= []<>Q"
wenzelm@21624
   826
  apply (erule leadsto_infinite [temp_use])
wenzelm@21624
   827
  apply (erule EnsuresLeadsto [temp_use])
wenzelm@21624
   828
  apply assumption
wenzelm@21624
   829
  done
wenzelm@21624
   830
wenzelm@21624
   831
wenzelm@21624
   832
(*** Gronning's lattice rules (taken from TLP) ***)
wenzelm@21624
   833
section "Lattice rules"
wenzelm@21624
   834
wenzelm@21624
   835
lemma LatticeReflexivity: "|- F ~> F"
wenzelm@21624
   836
  apply (unfold leadsto_def)
wenzelm@21624
   837
  apply (rule necT InitDmd_gen)+
wenzelm@21624
   838
  done
wenzelm@21624
   839
wenzelm@21624
   840
lemma LatticeTransitivity: "|- (G ~> H) & (F ~> G) --> (F ~> H)"
wenzelm@21624
   841
  apply (unfold leadsto_def)
wenzelm@21624
   842
  apply clarsimp
wenzelm@21624
   843
  apply (erule dup_boxE) (* [][] (Init G --> H) *)
wenzelm@42787
   844
  apply merge_box
wenzelm@21624
   845
  apply (clarsimp elim!: STL4E [temp_use])
wenzelm@21624
   846
  apply (rule dup_dmdD)
wenzelm@21624
   847
  apply (subgoal_tac "sigmaa |= <>Init G")
wenzelm@21624
   848
   apply (erule DmdImpl2)
wenzelm@21624
   849
   apply assumption
wenzelm@21624
   850
  apply (simp add: dmdInitD)
wenzelm@21624
   851
  done
wenzelm@21624
   852
wenzelm@21624
   853
lemma LatticeDisjunctionElim1: "|- (F | G ~> H) --> (F ~> H)"
wenzelm@21624
   854
  apply (unfold leadsto_def)
wenzelm@21624
   855
  apply (auto simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   856
  done
wenzelm@21624
   857
wenzelm@21624
   858
lemma LatticeDisjunctionElim2: "|- (F | G ~> H) --> (G ~> H)"
wenzelm@21624
   859
  apply (unfold leadsto_def)
wenzelm@21624
   860
  apply (auto simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   861
  done
wenzelm@21624
   862
wenzelm@21624
   863
lemma LatticeDisjunctionIntro: "|- (F ~> H) & (G ~> H) --> (F | G ~> H)"
wenzelm@21624
   864
  apply (unfold leadsto_def)
wenzelm@21624
   865
  apply clarsimp
wenzelm@42787
   866
  apply merge_box
wenzelm@21624
   867
  apply (auto simp: Init_simps elim!: STL4E [temp_use])
wenzelm@21624
   868
  done
wenzelm@21624
   869
wenzelm@21624
   870
lemma LatticeDisjunction: "|- (F | G ~> H) = ((F ~> H) & (G ~> H))"
wenzelm@21624
   871
  by (auto intro: LatticeDisjunctionIntro [temp_use]
wenzelm@21624
   872
    LatticeDisjunctionElim1 [temp_use]
wenzelm@21624
   873
    LatticeDisjunctionElim2 [temp_use])
wenzelm@21624
   874
wenzelm@21624
   875
lemma LatticeDiamond: "|- (A ~> B | C) & (B ~> D) & (C ~> D) --> (A ~> D)"
wenzelm@21624
   876
  apply clarsimp
wenzelm@21624
   877
  apply (subgoal_tac "sigma |= (B | C) ~> D")
wenzelm@21624
   878
  apply (erule_tac G = "LIFT (B | C)" in LatticeTransitivity [temp_use])
nipkow@44890
   879
   apply (fastforce intro!: LatticeDisjunctionIntro [temp_use])+
wenzelm@21624
   880
  done
wenzelm@21624
   881
wenzelm@21624
   882
lemma LatticeTriangle: "|- (A ~> D | B) & (B ~> D) --> (A ~> D)"
wenzelm@21624
   883
  apply clarsimp
wenzelm@21624
   884
  apply (subgoal_tac "sigma |= (D | B) ~> D")
wenzelm@21624
   885
   apply (erule_tac G = "LIFT (D | B)" in LatticeTransitivity [temp_use])
wenzelm@21624
   886
  apply assumption
wenzelm@21624
   887
  apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
wenzelm@21624
   888
  done
wenzelm@21624
   889
wenzelm@21624
   890
lemma LatticeTriangle2: "|- (A ~> B | D) & (B ~> D) --> (A ~> D)"
wenzelm@21624
   891
  apply clarsimp
wenzelm@21624
   892
  apply (subgoal_tac "sigma |= B | D ~> D")
wenzelm@21624
   893
   apply (erule_tac G = "LIFT (B | D)" in LatticeTransitivity [temp_use])
wenzelm@21624
   894
   apply assumption
wenzelm@21624
   895
  apply (auto intro: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use])
wenzelm@21624
   896
  done
wenzelm@21624
   897
wenzelm@21624
   898
(*** Lamport's fairness rules ***)
wenzelm@21624
   899
section "Fairness rules"
wenzelm@21624
   900
wenzelm@21624
   901
lemma WF1:
wenzelm@21624
   902
  "[| |- $P & N  --> P` | Q`;    
wenzelm@21624
   903
      |- ($P & N) & <A>_v --> Q`;    
wenzelm@21624
   904
      |- $P & N --> $(Enabled(<A>_v)) |]    
wenzelm@21624
   905
  ==> |- []N & WF(A)_v --> (P ~> Q)"
wenzelm@21624
   906
  apply (clarsimp dest!: BoxWFI [temp_use])
wenzelm@21624
   907
  apply (erule (2) ensures [temp_use])
wenzelm@21624
   908
  apply (erule (1) STL4Edup)
wenzelm@21624
   909
  apply (clarsimp simp: WF_def)
wenzelm@21624
   910
  apply (rule STL2 [temp_use])
wenzelm@21624
   911
  apply (clarsimp elim!: mp intro!: InitDmd [temp_use])
wenzelm@21624
   912
  apply (erule STL4 [temp_use, THEN box_stp_actD [temp_use]])
wenzelm@21624
   913
  apply (simp add: split_box_conj box_stp_actI)
wenzelm@21624
   914
  done
wenzelm@21624
   915
wenzelm@21624
   916
(* Sometimes easier to use; designed for action B rather than state predicate Q *)
wenzelm@21624
   917
lemma WF_leadsto:
wenzelm@21624
   918
  assumes 1: "|- N & $P --> $Enabled (<A>_v)"
wenzelm@21624
   919
    and 2: "|- N & <A>_v --> B"
wenzelm@21624
   920
    and 3: "|- [](N & [~A]_v) --> stable P"
wenzelm@21624
   921
  shows "|- []N & WF(A)_v --> (P ~> B)"
wenzelm@21624
   922
  apply (unfold leadsto_def)
wenzelm@21624
   923
  apply (clarsimp dest!: BoxWFI [temp_use])
wenzelm@21624
   924
  apply (erule (1) STL4Edup)
wenzelm@21624
   925
  apply clarsimp
wenzelm@21624
   926
  apply (rule 2 [THEN DmdImpl, temp_use])
wenzelm@21624
   927
  apply (rule BoxDmd_simple [temp_use])
wenzelm@21624
   928
   apply assumption
wenzelm@21624
   929
  apply (rule classical)
wenzelm@21624
   930
  apply (rule STL2 [temp_use])
wenzelm@21624
   931
  apply (clarsimp simp: WF_def elim!: mp intro!: InitDmd [temp_use])
wenzelm@21624
   932
  apply (rule 1 [THEN STL4, temp_use, THEN box_stp_actD])
wenzelm@21624
   933
  apply (simp (no_asm_simp) add: split_box_conj [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
   934
  apply (erule INV1 [temp_use])
wenzelm@21624
   935
  apply (rule 3 [temp_use])
wenzelm@21624
   936
  apply (simp add: split_box_conj [try_rewrite] NotDmd [temp_use] not_angle [try_rewrite])
wenzelm@21624
   937
  done
wenzelm@21624
   938
wenzelm@21624
   939
lemma SF1:
wenzelm@21624
   940
  "[| |- $P & N  --> P` | Q`;    
wenzelm@21624
   941
      |- ($P & N) & <A>_v --> Q`;    
wenzelm@21624
   942
      |- []P & []N & []F --> <>Enabled(<A>_v) |]    
wenzelm@21624
   943
  ==> |- []N & SF(A)_v & []F --> (P ~> Q)"
wenzelm@21624
   944
  apply (clarsimp dest!: BoxSFI [temp_use])
wenzelm@21624
   945
  apply (erule (2) ensures [temp_use])
wenzelm@21624
   946
  apply (erule_tac F = F in dup_boxE)
wenzelm@42787
   947
  apply merge_temp_box
wenzelm@21624
   948
  apply (erule STL4Edup)
wenzelm@21624
   949
  apply assumption
wenzelm@21624
   950
  apply (clarsimp simp: SF_def)
wenzelm@21624
   951
  apply (rule STL2 [temp_use])
wenzelm@21624
   952
  apply (erule mp)
wenzelm@21624
   953
  apply (erule STL4 [temp_use])
wenzelm@21624
   954
  apply (simp add: split_box_conj [try_rewrite] STL3 [try_rewrite])
wenzelm@21624
   955
  done
wenzelm@21624
   956
wenzelm@21624
   957
lemma WF2:
wenzelm@21624
   958
  assumes 1: "|- N & <B>_f --> <M>_g"
wenzelm@21624
   959
    and 2: "|- $P & P` & <N & A>_f --> B"
wenzelm@21624
   960
    and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"
wenzelm@21624
   961
    and 4: "|- [](N & [~B]_f) & WF(A)_f & []F & <>[]Enabled(<M>_g) --> <>[]P"
wenzelm@21624
   962
  shows "|- []N & WF(A)_f & []F --> WF(M)_g"
wenzelm@21624
   963
  apply (clarsimp dest!: BoxWFI [temp_use] BoxDmdBox [temp_use, THEN iffD2]
wenzelm@21624
   964
    simp: WF_def [where A = M])
wenzelm@21624
   965
  apply (erule_tac F = F in dup_boxE)
wenzelm@42787
   966
  apply merge_temp_box
wenzelm@21624
   967
  apply (erule STL4Edup)
wenzelm@21624
   968
   apply assumption
wenzelm@21624
   969
  apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
wenzelm@21624
   970
  apply (rule classical)
wenzelm@21624
   971
  apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")
wenzelm@21624
   972
   apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])
wenzelm@21624
   973
  apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])
wenzelm@21624
   974
  apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])
wenzelm@42787
   975
  apply merge_act_box
wenzelm@21624
   976
  apply (frule 4 [temp_use])
wenzelm@21624
   977
     apply assumption+
wenzelm@21624
   978
  apply (drule STL6 [temp_use])
wenzelm@21624
   979
   apply assumption
wenzelm@21624
   980
  apply (erule_tac V = "sigmaa |= <>[]P" in thin_rl)
wenzelm@21624
   981
  apply (erule_tac V = "sigmaa |= []F" in thin_rl)
wenzelm@21624
   982
  apply (drule BoxWFI [temp_use])
wenzelm@21624
   983
  apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)
wenzelm@42787
   984
  apply merge_temp_box
wenzelm@21624
   985
  apply (erule DmdImpldup)
wenzelm@21624
   986
   apply assumption
wenzelm@21624
   987
  apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]
wenzelm@21624
   988
    WF_Box [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
   989
   apply (force elim!: TLA2E [where P = P, temp_use])
wenzelm@21624
   990
  apply (rule STL2 [temp_use])
wenzelm@21624
   991
  apply (force simp: WF_def split_box_conj [try_rewrite]
wenzelm@21624
   992
    elim!: mp intro!: InitDmd [temp_use] 3 [THEN STL4, temp_use])
wenzelm@21624
   993
  done
wenzelm@21624
   994
wenzelm@21624
   995
lemma SF2:
wenzelm@21624
   996
  assumes 1: "|- N & <B>_f --> <M>_g"
wenzelm@21624
   997
    and 2: "|- $P & P` & <N & A>_f --> B"
wenzelm@21624
   998
    and 3: "|- P & Enabled(<M>_g) --> Enabled(<A>_f)"
wenzelm@21624
   999
    and 4: "|- [](N & [~B]_f) & SF(A)_f & []F & []<>Enabled(<M>_g) --> <>[]P"
wenzelm@21624
  1000
  shows "|- []N & SF(A)_f & []F --> SF(M)_g"
wenzelm@21624
  1001
  apply (clarsimp dest!: BoxSFI [temp_use] simp: 2 [try_rewrite] SF_def [where A = M])
wenzelm@21624
  1002
  apply (erule_tac F = F in dup_boxE)
wenzelm@21624
  1003
  apply (erule_tac F = "TEMP <>Enabled (<M>_g) " in dup_boxE)
wenzelm@42787
  1004
  apply merge_temp_box
wenzelm@21624
  1005
  apply (erule STL4Edup)
wenzelm@21624
  1006
   apply assumption
wenzelm@21624
  1007
  apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]])
wenzelm@21624
  1008
  apply (rule classical)
wenzelm@21624
  1009
  apply (subgoal_tac "sigmaa |= <> (($P & P` & N) & <A>_f)")
wenzelm@21624
  1010
   apply (force simp: angle_def intro!: 2 [temp_use] elim!: DmdImplE [temp_use])
wenzelm@21624
  1011
  apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use])
wenzelm@21624
  1012
  apply (simp add: NotDmd [temp_use] not_angle [try_rewrite])
wenzelm@42787
  1013
  apply merge_act_box
wenzelm@21624
  1014
  apply (frule 4 [temp_use])
wenzelm@21624
  1015
     apply assumption+
wenzelm@21624
  1016
  apply (erule_tac V = "sigmaa |= []F" in thin_rl)
wenzelm@21624
  1017
  apply (drule BoxSFI [temp_use])
wenzelm@21624
  1018
  apply (erule_tac F = "TEMP <>Enabled (<M>_g)" in dup_boxE)
wenzelm@21624
  1019
  apply (erule_tac F = "ACT N & [~B]_f" in dup_boxE)
wenzelm@42787
  1020
  apply merge_temp_box
wenzelm@21624
  1021
  apply (erule DmdImpldup)
wenzelm@21624
  1022
   apply assumption
wenzelm@21624
  1023
  apply (auto simp: split_box_conj [try_rewrite] STL3 [try_rewrite]
wenzelm@21624
  1024
    SF_Box [try_rewrite] box_stp_act [try_rewrite])
wenzelm@21624
  1025
   apply (force elim!: TLA2E [where P = P, temp_use])
wenzelm@21624
  1026
  apply (rule STL2 [temp_use])
wenzelm@21624
  1027
  apply (force simp: SF_def split_box_conj [try_rewrite]
wenzelm@21624
  1028
    elim!: mp InfImpl [temp_use] intro!: 3 [temp_use])
wenzelm@21624
  1029
  done
wenzelm@21624
  1030
wenzelm@21624
  1031
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1032
(***           Liveness proofs by well-founded orderings                   ***)
wenzelm@21624
  1033
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1034
section "Well-founded orderings"
wenzelm@21624
  1035
wenzelm@21624
  1036
lemma wf_leadsto:
wenzelm@21624
  1037
  assumes 1: "wf r"
wenzelm@21624
  1038
    and 2: "!!x. sigma |= F x ~> (G | (EX y. #((y,x):r) & F y))    "
wenzelm@21624
  1039
  shows "sigma |= F x ~> G"
wenzelm@21624
  1040
  apply (rule 1 [THEN wf_induct])
wenzelm@21624
  1041
  apply (rule LatticeTriangle [temp_use])
wenzelm@21624
  1042
   apply (rule 2)
wenzelm@21624
  1043
  apply (auto simp: leadsto_exists [try_rewrite])
wenzelm@21624
  1044
  apply (case_tac "(y,x) :r")
wenzelm@21624
  1045
   apply force
wenzelm@21624
  1046
  apply (force simp: leadsto_def Init_simps intro!: necT [temp_use])
wenzelm@21624
  1047
  done
wenzelm@21624
  1048
wenzelm@21624
  1049
(* If r is well-founded, state function v cannot decrease forever *)
wenzelm@21624
  1050
lemma wf_not_box_decrease: "!!r. wf r ==> |- [][ (v`, $v) : #r ]_v --> <>[][#False]_v"
wenzelm@21624
  1051
  apply clarsimp
wenzelm@21624
  1052
  apply (rule ccontr)
wenzelm@21624
  1053
  apply (subgoal_tac "sigma |= (EX x. v=#x) ~> #False")
wenzelm@21624
  1054
   apply (drule leadsto_false [temp_use, THEN iffD1, THEN STL2_gen [temp_use]])
wenzelm@21624
  1055
   apply (force simp: Init_defs)
wenzelm@21624
  1056
  apply (clarsimp simp: leadsto_exists [try_rewrite] not_square [try_rewrite] more_temp_simps)
wenzelm@21624
  1057
  apply (erule wf_leadsto)
wenzelm@21624
  1058
  apply (rule ensures_simple [temp_use])
wenzelm@21624
  1059
   apply (auto simp: square_def angle_def)
wenzelm@21624
  1060
  done
wenzelm@21624
  1061
wenzelm@21624
  1062
(* "wf r  ==>  |- <>[][ (v`, $v) : #r ]_v --> <>[][#False]_v" *)
wenzelm@21624
  1063
lemmas wf_not_dmd_box_decrease =
wenzelm@45605
  1064
  wf_not_box_decrease [THEN DmdImpl, unfolded more_temp_simps]
wenzelm@21624
  1065
wenzelm@21624
  1066
(* If there are infinitely many steps where v decreases, then there
wenzelm@21624
  1067
   have to be infinitely many non-stuttering steps where v doesn't decrease.
wenzelm@21624
  1068
*)
wenzelm@21624
  1069
lemma wf_box_dmd_decrease:
wenzelm@21624
  1070
  assumes 1: "wf r"
wenzelm@21624
  1071
  shows "|- []<>((v`, $v) : #r) --> []<><(v`, $v) ~: #r>_v"
wenzelm@21624
  1072
  apply clarsimp
wenzelm@21624
  1073
  apply (rule ccontr)
wenzelm@21624
  1074
  apply (simp add: not_angle [try_rewrite] more_temp_simps)
wenzelm@21624
  1075
  apply (drule 1 [THEN wf_not_dmd_box_decrease [temp_use]])
wenzelm@21624
  1076
  apply (drule BoxDmdDmdBox [temp_use])
wenzelm@21624
  1077
   apply assumption
wenzelm@21624
  1078
  apply (subgoal_tac "sigma |= []<> ((#False) ::action)")
wenzelm@21624
  1079
   apply force
wenzelm@21624
  1080
  apply (erule STL4E)
wenzelm@21624
  1081
  apply (rule DmdImpl)
wenzelm@21624
  1082
  apply (force intro: 1 [THEN wf_irrefl, temp_use])
wenzelm@21624
  1083
  done
wenzelm@21624
  1084
wenzelm@21624
  1085
(* In particular, for natural numbers, if n decreases infinitely often
wenzelm@21624
  1086
   then it has to increase infinitely often.
wenzelm@21624
  1087
*)
wenzelm@21624
  1088
lemma nat_box_dmd_decrease: "!!n::nat stfun. |- []<>(n` < $n) --> []<>($n < n`)"
wenzelm@21624
  1089
  apply clarsimp
wenzelm@21624
  1090
  apply (subgoal_tac "sigma |= []<><~ ((n`,$n) : #less_than) >_n")
wenzelm@21624
  1091
   apply (erule thin_rl)
wenzelm@21624
  1092
   apply (erule STL4E)
wenzelm@21624
  1093
   apply (rule DmdImpl)
wenzelm@21624
  1094
   apply (clarsimp simp: angle_def [try_rewrite])
wenzelm@21624
  1095
  apply (rule wf_box_dmd_decrease [temp_use])
wenzelm@21624
  1096
   apply (auto elim!: STL4E [temp_use] DmdImplE [temp_use])
wenzelm@21624
  1097
  done
wenzelm@21624
  1098
wenzelm@21624
  1099
wenzelm@21624
  1100
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1101
(***           Flexible quantification over state variables                ***)
wenzelm@21624
  1102
(* ------------------------------------------------------------------------- *)
wenzelm@21624
  1103
section "Flexible quantification"
wenzelm@21624
  1104
wenzelm@21624
  1105
lemma aallI:
wenzelm@21624
  1106
  assumes 1: "basevars vs"
wenzelm@21624
  1107
    and 2: "(!!x. basevars (x,vs) ==> sigma |= F x)"
wenzelm@21624
  1108
  shows "sigma |= (AALL x. F x)"
wenzelm@21624
  1109
  by (auto simp: aall_def elim!: eexE [temp_use] intro!: 1 dest!: 2 [temp_use])
wenzelm@21624
  1110
wenzelm@21624
  1111
lemma aallE: "|- (AALL x. F x) --> F x"
wenzelm@21624
  1112
  apply (unfold aall_def)
wenzelm@21624
  1113
  apply clarsimp
wenzelm@21624
  1114
  apply (erule contrapos_np)
wenzelm@21624
  1115
  apply (force intro!: eexI [temp_use])
wenzelm@21624
  1116
  done
wenzelm@21624
  1117
wenzelm@21624
  1118
(* monotonicity of quantification *)
wenzelm@21624
  1119
lemma eex_mono:
wenzelm@21624
  1120
  assumes 1: "sigma |= EEX x. F x"
wenzelm@21624
  1121
    and 2: "!!x. sigma |= F x --> G x"
wenzelm@21624
  1122
  shows "sigma |= EEX x. G x"
wenzelm@21624
  1123
  apply (rule unit_base [THEN 1 [THEN eexE]])
wenzelm@21624
  1124
  apply (rule eexI [temp_use])
wenzelm@21624
  1125
  apply (erule 2 [unfolded intensional_rews, THEN mp])
wenzelm@21624
  1126
  done
wenzelm@21624
  1127
wenzelm@21624
  1128
lemma aall_mono:
wenzelm@21624
  1129
  assumes 1: "sigma |= AALL x. F(x)"
wenzelm@21624
  1130
    and 2: "!!x. sigma |= F(x) --> G(x)"
wenzelm@21624
  1131
  shows "sigma |= AALL x. G(x)"
wenzelm@21624
  1132
  apply (rule unit_base [THEN aallI])
wenzelm@21624
  1133
  apply (rule 2 [unfolded intensional_rews, THEN mp])
wenzelm@21624
  1134
  apply (rule 1 [THEN aallE [temp_use]])
wenzelm@21624
  1135
  done
wenzelm@21624
  1136
wenzelm@21624
  1137
(* Derived history introduction rule *)
wenzelm@21624
  1138
lemma historyI:
wenzelm@21624
  1139
  assumes 1: "sigma |= Init I"
wenzelm@21624
  1140
    and 2: "sigma |= []N"
wenzelm@21624
  1141
    and 3: "basevars vs"
wenzelm@21624
  1142
    and 4: "!!h. basevars(h,vs) ==> |- I & h = ha --> HI h"
wenzelm@21624
  1143
    and 5: "!!h s t. [| basevars(h,vs); N (s,t); h t = hb (h s) (s,t) |] ==> HN h (s,t)"
wenzelm@21624
  1144
  shows "sigma |= EEX h. Init (HI h) & [](HN h)"
wenzelm@21624
  1145
  apply (rule history [temp_use, THEN eexE])
wenzelm@21624
  1146
  apply (rule 3)
wenzelm@21624
  1147
  apply (rule eexI [temp_use])
wenzelm@21624
  1148
  apply clarsimp
wenzelm@21624
  1149
  apply (rule conjI)
wenzelm@21624
  1150
   prefer 2
wenzelm@21624
  1151
   apply (insert 2)
wenzelm@42787
  1152
   apply merge_box
wenzelm@21624
  1153
   apply (force elim!: STL4E [temp_use] 5 [temp_use])
wenzelm@21624
  1154
  apply (insert 1)
wenzelm@21624
  1155
  apply (force simp: Init_defs elim!: 4 [temp_use])
wenzelm@21624
  1156
  done
wenzelm@21624
  1157
wenzelm@21624
  1158
(* ----------------------------------------------------------------------
wenzelm@21624
  1159
   example of a history variable: existence of a clock
wenzelm@21624
  1160
*)
wenzelm@21624
  1161
wenzelm@21624
  1162
lemma "|- EEX h. Init(h = #True) & [](h` = (~$h))"
wenzelm@21624
  1163
  apply (rule tempI)
wenzelm@21624
  1164
  apply (rule historyI)
wenzelm@21624
  1165
  apply (force simp: Init_defs intro!: unit_base [temp_use] necT [temp_use])+
wenzelm@21624
  1166
  done
wenzelm@21624
  1167
wenzelm@21624
  1168
end