Termination proof

1: switching to dependency pairs

a__eq#( s( X ) , s( Y ) )	→	a__eq#( X , Y )
mark#( eq( X1 , X2 ) )	→	a__eq#( X1 , X2 )
mark#( inf( X ) )	→	a__inf#( mark( X ) )
mark#( inf( X ) )	→	mark#( X )
mark#( take( X1 , X2 ) )	→	a__take#( mark( X1 ) , mark( X2 ) )
mark#( take( X1 , X2 ) )	→	mark#( X1 )
mark#( take( X1 , X2 ) )	→	mark#( X2 )
mark#( length( X ) )	→	a__length#( mark( X ) )
mark#( length( X ) )	→	mark#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( take( X1 , X2 ) )	→	mark#( X1 )
  mark#( inf( X ) )	→	mark#( X )
  mark#( take( X1 , X2 ) )	→	mark#( X2 )
  mark#( length( X ) )	→	mark#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	x1
  [length (x1) ]	=	x1
  [a__length (x1) ]	=	x1
  [a__eq (x1, x2) ]	=	0
  [eq (x1, x2) ]	=	0
  [mark# (x1) ]	=	2 x1
  [0]	=	0
  [nil]	=	1
  [cons (x1, x2) ]	=	x1
  [a__take (x1, x2) ]	=	x1 + 2 x2 + 2
  [true]	=	0
  [false]	=	0
  [s (x1) ]	=	0
  [take (x1, x2) ]	=	x1 + 2 x2 + 2
  [a__inf (x1) ]	=	2 x1
  [inf (x1) ]	=	2 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( inf( X ) )	→	mark#( X )
  mark#( length( X ) )	→	mark#( X )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	2 x1
  [length (x1) ]	=	2 x1
  [a__length (x1) ]	=	2 x1
  [a__eq (x1, x2) ]	=	2
  [eq (x1, x2) ]	=	1
  [mark# (x1) ]	=	x1
  [0]	=	0
  [nil]	=	0
  [cons (x1, x2) ]	=	0
  [a__take (x1, x2) ]	=	2
  [true]	=	0
  [false]	=	0
  [s (x1) ]	=	0
  [take (x1, x2) ]	=	2
  [a__inf (x1) ]	=	x1 + 1
  [inf (x1) ]	=	x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( length( X ) )

  →

  mark#( X )

  1.1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	2 x1
  [length (x1) ]	=	3 x1 + 2
  [a__length (x1) ]	=	3 x1 + 3
  [a__eq (x1, x2) ]	=	1
  [eq (x1, x2) ]	=	1
  [mark# (x1) ]	=	3 x1
  [0]	=	0
  [nil]	=	2
  [cons (x1, x2) ]	=	1
  [a__take (x1, x2) ]	=	2
  [true]	=	0
  [false]	=	0
  [s (x1) ]	=	2
  [take (x1, x2) ]	=	1
  [inf (x1) ]	=	2
  [a__inf (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: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  a__eq#( s( X ) , s( Y ) )

  →

  a__eq#( X , Y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	2 x1 + 1
  [length (x1) ]	=	0
  [a__length (x1) ]	=	1
  [a__eq (x1, x2) ]	=	3 x1 + 3 x2 + 2
  [eq (x1, x2) ]	=	2 x1 + 2 x2 + 2
  [0]	=	1
  [nil]	=	0
  [a__eq# (x1, x2) ]	=	x1
  [cons (x1, x2) ]	=	0
  [a__take (x1, x2) ]	=	0
  [true]	=	2
  [false]	=	0
  [s (x1) ]	=	x1 + 1
  [take (x1, x2) ]	=	0
  [inf (x1) ]	=	2
  [a__inf (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.