Termination proof

1: switching to dependency pairs

a__app#( nil , YS )	→	mark#( YS )
a__app#( cons( X , XS ) , YS )	→	mark#( X )
a__from#( X )	→	mark#( X )
a__zWadr#( cons( X , XS ) , cons( Y , YS ) )	→	a__app#( mark( Y ) , cons( mark( X ) , nil ) )
a__zWadr#( cons( X , XS ) , cons( Y , YS ) )	→	mark#( Y )
a__zWadr#( cons( X , XS ) , cons( Y , YS ) )	→	mark#( X )
mark#( app( X1 , X2 ) )	→	a__app#( mark( X1 ) , mark( X2 ) )
mark#( app( X1 , X2 ) )	→	mark#( X1 )
mark#( app( X1 , X2 ) )	→	mark#( X2 )
mark#( from( X ) )	→	a__from#( mark( X ) )
mark#( from( X ) )	→	mark#( X )
mark#( zWadr( X1 , X2 ) )	→	a__zWadr#( mark( X1 ) , mark( X2 ) )
mark#( zWadr( X1 , X2 ) )	→	mark#( X1 )
mark#( zWadr( X1 , X2 ) )	→	mark#( X2 )
mark#( prefix( X ) )	→	a__prefix#( mark( X ) )
mark#( prefix( X ) )	→	mark#( X )
mark#( cons( X1 , X2 ) )	→	mark#( X1 )
mark#( s( X ) )	→	mark#( X )

1.1: dependency graph processor

The dependency pairs are split into 1 component(s).

• The 1st component contains the pair(s)

  mark#( app( X1 , X2 ) )	→	a__app#( mark( X1 ) , mark( X2 ) )
  a__app#( nil , YS )	→	mark#( YS )
  mark#( app( X1 , X2 ) )	→	mark#( X1 )
  mark#( app( X1 , X2 ) )	→	mark#( X2 )
  mark#( from( X ) )	→	a__from#( mark( X ) )
  a__from#( X )	→	mark#( X )
  mark#( from( X ) )	→	mark#( X )
  mark#( zWadr( X1 , X2 ) )	→	a__zWadr#( mark( X1 ) , mark( X2 ) )
  a__zWadr#( cons( X , XS ) , cons( Y , YS ) )	→	a__app#( mark( Y ) , cons( mark( X ) , nil ) )
  a__app#( cons( X , XS ) , YS )	→	mark#( X )
  mark#( zWadr( X1 , X2 ) )	→	mark#( X1 )
  mark#( zWadr( X1 , X2 ) )	→	mark#( X2 )
  mark#( prefix( X ) )	→	mark#( X )
  mark#( cons( X1 , X2 ) )	→	mark#( X1 )
  mark#( s( X ) )	→	mark#( X )
  a__zWadr#( cons( X , XS ) , cons( Y , YS ) )	→	mark#( Y )
  a__zWadr#( cons( X , XS ) , cons( Y , YS ) )	→	mark#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2 x1
  [a__app (x1, x2) ]	=	x1 + 2 x2 + 2
  [a__zWadr (x1, x2) ]	=	2 x1 + 2 x2 + 2
  [mark (x1) ]	=	x1
  [a__prefix (x1) ]	=	3 x1 + 3
  [a__app# (x1, x2) ]	=	2 x1 + 2 x2
  [mark# (x1) ]	=	2 x1
  [nil]	=	0
  [zWadr (x1, x2) ]	=	2 x1 + 2 x2 + 2
  [cons (x1, x2) ]	=	x1
  [app (x1, x2) ]	=	x1 + 2 x2 + 2
  [a__from (x1) ]	=	2 x1
  [s (x1) ]	=	2 x1 + 3
  [a__zWadr# (x1, x2) ]	=	2 x1 + 2 x2 + 2
  [a__from# (x1) ]	=	2 x1
  [prefix (x1) ]	=	3 x1 + 3
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  a__app#( nil , YS )	→	mark#( YS )
  mark#( from( X ) )	→	a__from#( mark( X ) )
  a__from#( X )	→	mark#( X )
  mark#( from( X ) )	→	mark#( X )
  a__app#( cons( X , XS ) , YS )	→	mark#( X )
  mark#( cons( X1 , X2 ) )	→	mark#( X1 )

  1.1.1.1: dependency graph processor

  The dependency pairs are split into 1 component(s).

  • The 1st component contains the pair(s)

    a__from#( X )	→	mark#( X )
    mark#( from( X ) )	→	a__from#( mark( X ) )
    mark#( from( X ) )	→	mark#( X )
    mark#( cons( X1 , X2 ) )	→	mark#( X1 )

    1.1.1.1.1: reduction pair processor

    Using the following reduction pair

    Linear polynomial interpretation over the naturals

    [from (x1) ]	=	x1 + 1
    [a__app (x1, x2) ]	=	2 x1 + 2 x2
    [mark (x1) ]	=	x1
    [a__zWadr (x1, x2) ]	=	2 x1 + 2 x2
    [a__prefix (x1) ]	=	1
    [mark# (x1) ]	=	x1
    [nil]	=	1
    [zWadr (x1, x2) ]	=	2 x1 + 2 x2
    [cons (x1, x2) ]	=	x1
    [app (x1, x2) ]	=	2 x1 + 2 x2
    [a__from (x1) ]	=	x1 + 1
    [a__from# (x1) ]	=	x1
    [s (x1) ]	=	0
    [prefix (x1) ]	=	1
    [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

    one remains with the following pair(s).

    a__from#( X )	→	mark#( X )
    mark#( cons( X1 , X2 ) )	→	mark#( X1 )

    1.1.1.1.1.1: dependency graph processor

    The dependency pairs are split into 1 component(s).

    • The 1st component contains the pair(s)

      mark#( cons( X1 , X2 ) )

      →

      mark#( X1 )

      1.1.1.1.1.1.1: reduction pair processor

      Using the following reduction pair

      Linear polynomial interpretation over the naturals

      [a__app (x1, x2) ]	=	x1 + x2
      [from (x1) ]	=	2 x1 + 1
      [mark (x1) ]	=	x1
      [a__zWadr (x1, x2) ]	=	2 x1 + 2 x2
      [a__prefix (x1) ]	=	2
      [mark# (x1) ]	=	2 x1
      [nil]	=	0
      [zWadr (x1, x2) ]	=	2 x1 + 2 x2
      [cons (x1, x2) ]	=	x1 + 1
      [app (x1, x2) ]	=	x1 + x2
      [a__from (x1) ]	=	2 x1 + 1
      [s (x1) ]	=	0
      [prefix (x1) ]	=	2
      [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

      one remains with the following pair(s).

      none

      1.1.1.1.1.1.1.1: P is empty

      All dependency pairs have been removed.