Theory ShortExecutions

(*  Title:      HOLCF/IOA/meta_theory/ShortExecutions.thy
    ID:         $Id: ShortExecutions.thy,v 1.10 2005/09/03 20:27:06 wenzelm Exp $
    Author:     Olaf Müller
*)

theory ShortExecutions
imports Traces
begin

text {*
  Some properties about @{text "Cut ex"}, defined as follows:

  For every execution ex there is another shorter execution @{text "Cut ex"}
  that has the same trace as ex, but its schedule ends with an external action.
*}

consts

(*  LastActExtex      ::"('a,'s)ioa => ('a,'s) pairs  => bool"*)
  LastActExtsch     ::"('a,'s)ioa => 'a Seq         => bool"

  Cut               ::"('a => bool) => 'a Seq    => 'a Seq"

  oraclebuild       ::"('a => bool) => 'a Seq -> 'a Seq -> 'a Seq"

defs

LastActExtsch_def:
  "LastActExtsch A sch == (Cut (%x. x: ext A) sch = sch)"

(* LastActExtex_def:
  "LastActExtex A ex == LastActExtsch A (filter_act$ex)" *)

Cut_def:
  "Cut P s == oraclebuild P$s$(Filter P$s)"

oraclebuild_def:
  "oraclebuild P == (fix$(LAM h s t. case t of
       nil => nil
    | x##xs =>
      (case x of
        UU => UU
      | Def y => (Takewhile (%x.~P x)$s)
                  @@ (y>>(h$(TL$(Dropwhile (%x.~ P x)$s))$xs))
      )
    ))"


axioms

Cut_prefixcl_Finite:
  "Finite s ==> (? y. s = Cut P s @@ y)"

LastActExtsmall1:
  "LastActExtsch A sch ==> LastActExtsch A (TL$(Dropwhile P$sch))"

LastActExtsmall2:
  "[| Finite sch1; LastActExtsch A (sch1 @@ sch2) |] ==> LastActExtsch A sch2"

ML {* use_legacy_bindings (the_context ()) *}

end

oraclebuild rewrite rules

theorem oraclebuild_unfold:

  oraclebuild P =
  (LAM s t.
      case t of nil => nil
      | x ## xs =>
          case x of UU => UU
          | Def y =>
              Takewhile (%a. ¬ P a)·s @@
              y>>oraclebuild P·(TL·(Dropwhile (%a. ¬ P a)·s))·xs)

theorem oraclebuild_UU:

  oraclebuild P·sch·UU = UU

theorem oraclebuild_nil:

  oraclebuild P·sch·nil = nil

theorem oraclebuild_cons:

  oraclebuild P·s·(x>>t) =
  Takewhile (%a. ¬ P a)·s @@ x>>oraclebuild P·(TL·(Dropwhile (%a. ¬ P a)·s))·t

Cut rewrite rules

theorem Cut_nil:

  [| Forall (%a. ¬ P a) s; Finite s |] ==> Cut P s = nil

theorem Cut_UU:

  [| Forall (%a. ¬ P a) s; ¬ Finite s |] ==> Cut P s = UU

theorem Cut_Cons:

  [| P t; Forall (%x. ¬ P x) ss; Finite ss |]
  ==> Cut P (ss @@ t>>rs) = ss @@ t>>Cut P rs

Cut lemmas for main theorem

theorem FilterCut:

  Filter P·s = Filter P·(Cut P s)

theorem Cut_idemp:

  Cut P (Cut P s) = Cut P s

theorem MapCut:

  Map f·(Cut (P o f) s) = Cut P (Map f·s)

theorem Cut_prefixcl_nFinite:

  ¬ Finite s ==> Cut P s << s

theorem execThruCut:

  is_exec_frag A (s, ex) ==> is_exec_frag A (s, Cut P ex)

Main Cut Theorem

theorem exists_LastActExtsch:

  [| sch ∈ schedules A; tr = Filter (%a. a ∈ ext A)·sch |]
  ==> ∃sch. sch ∈ schedules A ∧
            tr = Filter (%a. a ∈ ext A)·sch ∧ LastActExtsch A sch

Further Cut lemmas

theorem LastActExtimplnil:

  [| LastActExtsch A sch; Filter (%x. x ∈ ext A)·sch = nil |] ==> sch = nil

theorem LastActExtimplUU:

  [| LastActExtsch A sch; Filter (%x. x ∈ ext A)·sch = UU |] ==> sch = UU