Termination proof

1: switching to dependency pairs

a__fst#( s( X ) , cons( Y , Z ) )	→	mark#( Y )
a__from#( X )	→	mark#( X )
a__add#( 0 , X )	→	mark#( X )
mark#( fst( X1 , X2 ) )	→	a__fst#( mark( X1 ) , mark( X2 ) )
mark#( fst( X1 , X2 ) )	→	mark#( X1 )
mark#( fst( X1 , X2 ) )	→	mark#( X2 )
mark#( from( X ) )	→	a__from#( mark( X ) )
mark#( from( X ) )	→	mark#( X )
mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , mark( X2 ) )
mark#( add( X1 , X2 ) )	→	mark#( X1 )
mark#( add( X1 , X2 ) )	→	mark#( X2 )
mark#( len( X ) )	→	a__len#( mark( X ) )
mark#( len( X ) )	→	mark#( X )
mark#( cons( X1 , X2 ) )	→	mark#( X1 )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( fst( X1 , X2 ) )	→	a__fst#( mark( X1 ) , mark( X2 ) )
  a__fst#( s( X ) , cons( Y , Z ) )	→	mark#( Y )
  mark#( fst( X1 , X2 ) )	→	mark#( X1 )
  mark#( fst( X1 , X2 ) )	→	mark#( X2 )
  mark#( from( X ) )	→	a__from#( mark( X ) )
  a__from#( X )	→	mark#( X )
  mark#( from( X ) )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , mark( X2 ) )
  a__add#( 0 , X )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	mark#( X1 )
  mark#( add( X1 , X2 ) )	→	mark#( X2 )
  mark#( len( X ) )	→	mark#( X )
  mark#( cons( X1 , X2 ) )	→	mark#( X1 )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	3 x1 + 2
  [fst (x1, x2) ]	=	x1 + 2 x2
  [mark (x1) ]	=	2 x1
  [mark# (x1) ]	=	x1 + 1
  [0]	=	0
  [a__add# (x1, x2) ]	=	x1 + 2
  [a__add (x1, x2) ]	=	3 x1 + 3 x2 + 3
  [nil]	=	0
  [cons (x1, x2) ]	=	x1 + 2
  [a__fst# (x1, x2) ]	=	x1
  [len (x1) ]	=	2 x1 + 3
  [a__fst (x1, x2) ]	=	x1 + 2 x2
  [add (x1, x2) ]	=	3 x1 + 3 x2 + 3
  [a__from (x1) ]	=	3 x1 + 3
  [a__len (x1) ]	=	2 x1 + 3
  [a__from# (x1) ]	=	x1 + 2
  [s (x1) ]	=	0
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( fst( X1 , X2 ) )	→	mark#( X1 )
  mark#( fst( X1 , X2 ) )	→	mark#( X2 )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2
  [fst (x1, x2) ]	=	2 x1 + 2 x2 + 1
  [mark (x1) ]	=	x1
  [mark# (x1) ]	=	x1
  [0]	=	0
  [a__add (x1, x2) ]	=	2 x1 + 3
  [nil]	=	0
  [cons (x1, x2) ]	=	2
  [len (x1) ]	=	0
  [a__fst (x1, x2) ]	=	2 x1 + 2 x2 + 1
  [add (x1, x2) ]	=	2 x1 + 3
  [a__from (x1) ]	=	2
  [a__len (x1) ]	=	0
  [s (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: P is empty

  All dependency pairs have been removed.