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products/sources/formale Sprachen/PVS/interval_arith/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 28.9.2014 mit Größe 5 kB

Quelle numerical_bandb.pvs Sprache: PVS

numerical_bandb : THEORY
BEGIN

  IMPORTING safe_arith,
            interval_expr,
            structures@array2list[real],
            structures@Maybe,
            reals@real_orders

  IntervalMinMax : TYPE = [#
    mm     : Interval,
    lb_max : real,
    lb_box : ProperBox,
    ub_min : real,
    ub_box : ProperBox
  #]

  IMPORTING structures@branch_and_bound[RealExpr,IntervalMinMax,ProperBox,nat]

  v         : VAR nat
  maxdepth  : VAR nat
  expr      : VAR RealExpr
  box       : VAR ProperBox
  dirvar    : VAR DirVar
  dirvars   : VAR DirVarStack
  M,M1,M2   : VAR IntervalMinMax
  out       : VAR Output
  precision : VAR posreal
  prune,ge  : VAR ExitPred
  le        : VAR LocalExitPred
  select    : VAR DirVarSelector
  acc       : VAR IntervalMinMax

  evaluate(box,expr) : IntervalMinMax =
    LET X   = Eval(expr,box),
        Lb  = Lbbox(box),
        Ub  = Ubbox(box),
        P1  = Eval(expr,Lb),
        P2  = Eval(expr,Ub),
        (lb_max,lb_box) = IF ub(P1) < ub(P2) THEN (ub(P1),Lb) ELSE (ub(P2),Ub) ENDIF,
        (ub_min,ub_box) = IF lb(P1) > lb(P2) THEN (lb(P1),Lb) ELSE (lb(P2),Ub) ENDIF IN
    (# mm     := X,
       lb_max := lb_max,
       lb_box := lb_box,
       ub_min := ub_min,
       ub_box := ub_box
    #)

  branch(v,expr) : [RealExpr,RealExpr] = (expr,expr)
  denorm(dirvar,M) : IntervalMinMax = M

  combine(v,M1,M2) : IntervalMinMax =
    LET M_lb = IF M1`lb_max < M2`lb_max THEN M1 ELSE M2 ENDIF,
        M_ub = IF M1`ub_min > M2`ub_min THEN M1 ELSE M2 ENDIF IN
    (# mm     := Safe2(Union)(mm(M1),mm(M2)),
       lb_max := M_lb`lb_max,
       lb_box := M_lb`lb_box,
       ub_min := M_ub`ub_min,
       ub_box := M_ub`ub_box #)

  % <= 0 : min
  % >= 0 : max
  min_or_max: VAR int

  prune_mm(min_or_max)(dirvars,acc,M) : bool =
    (min_or_max <= 0 IMPLIES acc`lb_max <= lb(M`mm)) AND
    (min_or_max >= 0 IMPLIES ub(M`mm) <= acc`ub_min)

  le_mm(min_or_max,precision)(M) : bool =
    (min_or_max <= 0 IMPLIES LET lb = M`lb_max - lb(M`mm) IN lb <= precision) AND
    (min_or_max >= 0 IMPLIES LET ub = ub(M`mm) - M`ub_min IN ub <= precision)

  ge_mm(dirvars,acc,M) : bool = EmptyInterval?(M`mm)

  accumulate(acc,M) : IntervalMinMax =
    combine(0,acc,M)

  subdivide(v,box) : [ProperBox,ProperBox] = split(v,box)

  VarSel : TYPE = [# thisvar: nat, lb: real, ub: real, lbub: Interval #]

  var_varsel(expr:RealExpr,box:ProperBox,both:bool,
             Mb:listn[Interval](length(box)))
            (v:below(length(box))) : VarSel =
    LET X    = nth(box,v),
     lb   = IF both THEN lb(Eval(expr,setnth(Mb,v,LAMBDA(x:Interval): [| lb(X) |]))) ELSE 0 ENDIF,
     ub   = IF both THEN ub(Eval(expr,setnth(Mb,v,LAMBDA(x:Interval): [| ub(X) |]))) ELSE 0 ENDIF,
lbub = Eval(expr,setnth(Mb,v,LAMBDA(x:Interval):X)) IN
    (# thisvar  := v,
       lb       := lb,
       ub       := ub,
       lbub     := lbub
     #)

  max_varsel(vs1,vs2:VarSel) : VarSel =
    IF size(vs1`lbub) > size(vs2`lbub) THEN vs1
    ELSE vs2
    ENDIF

  IMPORTING structures@for_iterate

  both, dir : VAR bool

  mindir_maxvar_aux(both:bool)(dirvars,acc,box,expr) : DirVar =
    IF length(box) <= 1 THEN left(0)
    ELSE
      LET Mb = Midbox(box),
          vs = iterate_left(0,length(box)-1,var_varsel(expr,box,both,Mb),max_varsel) IN
      (vs`lb < vs`ub,vs`thisvar)
    ENDIF

  mindir_maxvar(dirvars,acc,box,expr) : DirVar =
    mindir_maxvar_aux(TRUE)(dirvars,acc,box,expr)

  maxdir_maxvar(dirvars,acc,box,expr) : DirVar =
    LET (dir,v) = mindir_maxvar(dirvars,acc,box,expr) IN
    (NOT dir,v)

  altdir_maxvar(dirvars,acc,box,expr) : DirVar =
    LET (dir,v) = mindir_maxvar_aux(FALSE)(dirvars,acc,box,expr) IN
      (even?(length(dirvars)),v)

  dir_maxvar(dir)(dirvars,acc,box,expr) : DirVar =
    LET v = mindir_maxvar_aux(FALSE)(dirvars,acc,box,expr)`2 IN
      (dir,v)

  max_rec(n:posnat,m:real,v:below(n),
                 i:subrange(1,n),b:Box | n = length(b)+i) : RECURSIVE below(n) =
    IF null?(b) THEN v
    ELSE LET mm = size(car(b)) IN
           IF mm > m THEN max_rec(n,mm,i,i+1,cdr(b))
           ELSE max_rec(n,m,v,i+1,cdr(b))
           ENDIF
    ENDIF
  MEASURE b BY <<

  max_aux(box) : nat =
    IF length(box) <= 1 THEN 0
    ELSE max_rec(length(box),size(car(box)),0,1,cdr(box))
    ENDIF

  dir_max(dir)(dirvars,acc,box,expr) : DirVar =
    (dir,max_aux(box))

  alt_max(dirvars,acc,box,expr) : DirVar =
    (even?(length(dirvars)),max_aux(box))

  altvar(dirvars,box) : nat =
    IF length(box) <= 1 THEN 0
    ELSE mod(length(dirvars),length(box))
    ENDIF

  dir_alt(dir)(dirvars,acc,box,expr) : DirVar =
    (dir,altvar(dirvars,box))

  alt_alt(dirvars,acc,box,expr) : DirVar =
    (even?(length(dirvars)),altvar(dirvars,box))

  interval_minmax(maxdepth,select,le,ge,prune)(expr,box) : Output =
    b_and_b_id(evaluate,branch,subdivide,denorm,combine,prune,le,ge,
               select,accumulate,maxdepth,expr,box)

  VVars(dirvars,box) : nat =
    IF length(box) <= 1 THEN 0
    ELSE mod(length(dirvars),4)
    ENDIF

  VMZ(dirvars,acc,box,expr) : DirVar =
    (True,VVars(dirvars,box))

  sound?(box,expr,M) : bool =
    Proper?(M`mm) IMPLIES
      (FORALL (vs:(vars_in_box?(box))): eval(expr,vs,length(box)) ## M`mm) AND
      Inclusion?(M`lb_box,box) AND
      Inclusion?(M`ub_box,box) AND
      (FORALL (vs:(vars_in_box?(M`lb_box))): eval(expr,vs,length(box)) <= M`lb_max) AND
      (FORALL (vs:(vars_in_box?(M`ub_box))): eval(expr,vs,length(box)) >= M`ub_min)

  interval_minmax_soundness : THEOREM
    sound?(box,expr,interval_minmax(maxdepth,select,le,ge,prune)(expr,box)`ans)

  numerical(maxdepth,precision,select,min_or_max)(expr,box) : Output =
    interval_minmax(maxdepth,select,le_mm(min_or_max,precision),ge_mm,prune_mm(min_or_max))(expr,box)

  numerical_soundness : THEOREM
    sound?(box,expr,numerical(maxdepth,precision,select,min_or_max)(expr,box)`ans)

END numerical_bandb

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