Termination proof

1: switching to dependency pairs

active#( f( g( X ) , Y ) )	→	mark#( f( X , f( g( X ) , Y ) ) )
active#( f( g( X ) , Y ) )	→	f#( X , f( g( X ) , Y ) )
active#( f( g( X ) , Y ) )	→	f#( g( X ) , Y )
active#( f( g( X ) , Y ) )	→	g#( X )
mark#( f( X1 , X2 ) )	→	active#( f( mark( X1 ) , X2 ) )
mark#( f( X1 , X2 ) )	→	f#( mark( X1 ) , X2 )
mark#( f( X1 , X2 ) )	→	mark#( X1 )
mark#( g( X ) )	→	active#( g( mark( X ) ) )
mark#( g( X ) )	→	g#( mark( X ) )
mark#( g( X ) )	→	mark#( X )
f#( mark( X1 ) , X2 )	→	f#( X1 , X2 )
f#( X1 , mark( X2 ) )	→	f#( X1 , X2 )
f#( active( X1 ) , X2 )	→	f#( X1 , X2 )
f#( X1 , active( X2 ) )	→	f#( X1 , X2 )
g#( mark( X ) )	→	g#( X )
g#( active( X ) )	→	g#( X )

1.1: dependency graph processor

The dependency pairs are split into 3 component(s).

• The 1st component contains the pair(s)

  mark#( f( X1 , X2 ) )	→	active#( f( mark( X1 ) , X2 ) )
  active#( f( g( X ) , Y ) )	→	mark#( f( X , f( g( X ) , Y ) ) )
  mark#( f( X1 , X2 ) )	→	mark#( X1 )
  mark#( g( X ) )	→	active#( g( mark( X ) ) )
  mark#( g( X ) )	→	mark#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	x1
  [active# (x1) ]	=	x1
  [f (x1, x2) ]	=	2 x1 + 2
  [active (x1) ]	=	x1
  [mark# (x1) ]	=	x1
  [g (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( f( X1 , X2 ) )	→	active#( f( mark( X1 ) , X2 ) )
  active#( f( g( X ) , Y ) )	→	mark#( f( X , f( g( X ) , Y ) ) )
  mark#( g( X ) )	→	active#( g( mark( X ) ) )
  mark#( g( X ) )	→	mark#( X )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	x1
  [active# (x1) ]	=	2 x1
  [f (x1, x2) ]	=	2 x1
  [active (x1) ]	=	x1
  [mark# (x1) ]	=	2 x1 + 1
  [g (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).

  none

  1.1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  f#( X1 , mark( X2 ) )	→	f#( X1 , X2 )
  f#( mark( X1 ) , X2 )	→	f#( X1 , X2 )
  f#( active( X1 ) , X2 )	→	f#( X1 , X2 )
  f#( X1 , active( X2 ) )	→	f#( X1 , X2 )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	2 x1 + 1
  [active (x1) ]	=	x1 + 3
  [f (x1, x2) ]	=	2
  [f# (x1, x2) ]	=	3 x1 + x2
  [g (x1) ]	=	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.

• The 3rd component contains the pair(s)

  g#( active( X ) )	→	g#( X )
  g#( mark( X ) )	→	g#( X )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [g# (x1) ]	=	x1
  [mark (x1) ]	=	x1 + 3
  [active (x1) ]	=	x1 + 3
  [f (x1, x2) ]	=	1
  [g (x1) ]	=	0
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.3.1: P is empty

  All dependency pairs have been removed.