Termination proof

1: switching to dependency pairs

a____#( __( X , Y ) , Z )	→	a____#( mark( X ) , a____( mark( Y ) , mark( Z ) ) )
a____#( __( X , Y ) , Z )	→	mark#( X )
a____#( __( X , Y ) , Z )	→	a____#( mark( Y ) , mark( Z ) )
a____#( __( X , Y ) , Z )	→	mark#( Y )
a____#( __( X , Y ) , Z )	→	mark#( Z )
a____#( X , nil )	→	mark#( X )
a____#( nil , X )	→	mark#( X )
a__U11#( tt )	→	a__U12#( tt )
a__isNePal#( __( I , __( P , I ) ) )	→	a__U11#( tt )
mark#( __( X1 , X2 ) )	→	a____#( mark( X1 ) , mark( X2 ) )
mark#( __( X1 , X2 ) )	→	mark#( X1 )
mark#( __( X1 , X2 ) )	→	mark#( X2 )
mark#( U11( X ) )	→	a__U11#( mark( X ) )
mark#( U11( X ) )	→	mark#( X )
mark#( U12( X ) )	→	a__U12#( mark( X ) )
mark#( U12( X ) )	→	mark#( X )
mark#( isNePal( X ) )	→	a__isNePal#( mark( X ) )
mark#( isNePal( X ) )	→	mark#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  a____#( __( X , Y ) , Z )	→	mark#( X )
  mark#( __( X1 , X2 ) )	→	a____#( mark( X1 ) , mark( X2 ) )
  a____#( __( X , Y ) , Z )	→	a____#( mark( X ) , a____( mark( Y ) , mark( Z ) ) )
  a____#( __( X , Y ) , Z )	→	a____#( mark( Y ) , mark( Z ) )
  a____#( __( X , Y ) , Z )	→	mark#( Y )
  mark#( __( X1 , X2 ) )	→	mark#( X1 )
  mark#( __( X1 , X2 ) )	→	mark#( X2 )
  mark#( U11( X ) )	→	mark#( X )
  mark#( U12( X ) )	→	mark#( X )
  mark#( isNePal( X ) )	→	mark#( X )
  a____#( __( X , Y ) , Z )	→	mark#( Z )
  a____#( X , nil )	→	mark#( X )
  a____#( nil , X )	→	mark#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [__ (x1, x2) ]	=	x1 + x2
  [mark (x1) ]	=	x1
  [isNePal (x1) ]	=	x1
  [a____ (x1, x2) ]	=	x1 + x2
  [mark# (x1) ]	=	x1
  [nil]	=	2
  [tt]	=	0
  [a__U11 (x1) ]	=	2 x1
  [a__isNePal (x1) ]	=	x1
  [a__U12 (x1) ]	=	x1
  [U11 (x1) ]	=	2 x1
  [a____# (x1, x2) ]	=	x1 + x2
  [U12 (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  a____#( __( X , Y ) , Z )	→	mark#( X )
  mark#( __( X1 , X2 ) )	→	a____#( mark( X1 ) , mark( X2 ) )
  a____#( __( X , Y ) , Z )	→	a____#( mark( X ) , a____( mark( Y ) , mark( Z ) ) )
  a____#( __( X , Y ) , Z )	→	a____#( mark( Y ) , mark( Z ) )
  a____#( __( X , Y ) , Z )	→	mark#( Y )
  mark#( __( X1 , X2 ) )	→	mark#( X1 )
  mark#( __( X1 , X2 ) )	→	mark#( X2 )
  mark#( U11( X ) )	→	mark#( X )
  mark#( U12( X ) )	→	mark#( X )
  mark#( isNePal( X ) )	→	mark#( X )
  a____#( __( X , Y ) , Z )	→	mark#( Z )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [__ (x1, x2) ]	=	x1 + x2 + 3
  [mark (x1) ]	=	x1
  [isNePal (x1) ]	=	2 x1
  [a____ (x1, x2) ]	=	x1 + x2 + 3
  [mark# (x1) ]	=	x1 + 2
  [nil]	=	0
  [tt]	=	3
  [a__U11 (x1) ]	=	2 x1 + 1
  [a__isNePal (x1) ]	=	2 x1
  [a__U12 (x1) ]	=	2 x1 + 1
  [U11 (x1) ]	=	2 x1 + 1
  [a____# (x1, x2) ]	=	x1 + x2 + 1
  [U12 (x1) ]	=	2 x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  a____#( __( X , Y ) , Z )	→	a____#( mark( X ) , a____( mark( Y ) , mark( Z ) ) )
  mark#( isNePal( X ) )	→	mark#( X )

  1.1.1.1.1: dependency graph processor

  The dependency pairs are split into 2 component(s).

  • The 1st component contains the pair(s)

    a____#( __( X , Y ) , Z )

    →

    a____#( mark( X ) , a____( mark( Y ) , mark( Z ) ) )

    1.1.1.1.1.1: reduction pair processor

    Using the following reduction pair

    Linear polynomial interpretation over the naturals

    [__ (x1, x2) ]	=	2 x1 + x2 + 1
    [mark (x1) ]	=	x1
    [a__U11 (x1) ]	=	0
    [a__isNePal (x1) ]	=	0
    [isNePal (x1) ]	=	0
    [a____ (x1, x2) ]	=	2 x1 + x2 + 1
    [a__U12 (x1) ]	=	0
    [U11 (x1) ]	=	0
    [a____# (x1, x2) ]	=	x1
    [nil]	=	0
    [tt]	=	0
    [U12 (x1) ]	=	0
    [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: P is empty

    All dependency pairs have been removed.

  • The 2nd component contains the pair(s)

    mark#( isNePal( X ) )

    →

    mark#( X )

    1.1.1.1.1.2: reduction pair processor

    Using the following reduction pair

    Linear polynomial interpretation over the naturals

    [__ (x1, x2) ]	=	2 x1 + x2
    [mark (x1) ]	=	x1
    [a__U11 (x1) ]	=	0
    [isNePal (x1) ]	=	2 x1 + 2
    [a__isNePal (x1) ]	=	2 x1 + 2
    [a____ (x1, x2) ]	=	2 x1 + x2
    [mark# (x1) ]	=	3 x1
    [a__U12 (x1) ]	=	0
    [U11 (x1) ]	=	0
    [nil]	=	2
    [tt]	=	0
    [U12 (x1) ]	=	0
    [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

    one remains with the following pair(s).

    none

    1.1.1.1.1.2.1: P is empty

    All dependency pairs have been removed.