Termination proof

1: switching to dependency pairs

a__f#( a , X , X )	→	a__f#( X , a__b , b )
a__f#( a , X , X )	→	a__b#
mark#( f( X1 , X2 , X3 ) )	→	a__f#( X1 , mark( X2 ) , X3 )
mark#( f( X1 , X2 , X3 ) )	→	mark#( X2 )
mark#( b )	→	a__b#

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( f( X1 , X2 , X3 ) )

  →

  mark#( X2 )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [b]	=	0
  [a]	=	0
  [mark (x1) ]	=	x1
  [f (x1, x2, x3) ]	=	2 x1 + 2
  [mark# (x1) ]	=	3 x1
  [a__f (x1, x2, x3) ]	=	2 x1 + 2
  [a__b]	=	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: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  a__f#( a , X , X )

  →

  a__f#( X , a__b , b )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [b]	=	0
  [a]	=	3
  [mark (x1) ]	=	2 x1 + 3
  [f (x1, x2, x3) ]	=	x1
  [a__f# (x1, x2, x3) ]	=	3 x1 + 2 x2 + 2 x3
  [a__f (x1, x2, x3) ]	=	2 x1 + 2
  [a__b]	=	3
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.2.1: P is empty

  All dependency pairs have been removed.