Termination proof

1: switching to dependency pairs

a__filter#( cons( X , Y ) , s( N ) , M )	→	mark#( X )
a__sieve#( cons( s( N ) , Y ) )	→	mark#( N )
a__nats#( N )	→	mark#( N )
a__zprimes#	→	a__sieve#( a__nats( s( s( 0 ) ) ) )
a__zprimes#	→	a__nats#( s( s( 0 ) ) )
mark#( filter( X1 , X2 , X3 ) )	→	a__filter#( mark( X1 ) , mark( X2 ) , mark( X3 ) )
mark#( filter( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( filter( X1 , X2 , X3 ) )	→	mark#( X2 )
mark#( filter( X1 , X2 , X3 ) )	→	mark#( X3 )
mark#( sieve( X ) )	→	a__sieve#( mark( X ) )
mark#( sieve( X ) )	→	mark#( X )
mark#( nats( X ) )	→	a__nats#( mark( X ) )
mark#( nats( X ) )	→	mark#( X )
mark#( zprimes )	→	a__zprimes#
mark#( cons( X1 , X2 ) )	→	mark#( X1 )
mark#( s( X ) )	→	mark#( X )

1.1: reduction pair processor

Linear polynomial interpretation over the naturals

[a__nats# (x1) ]	=	2 x1
[filter (x1, x2, x3) ]	=	x1 + x2 + x3 + 2
[mark (x1) ]	=	x1
[sieve (x1) ]	=	2 x1 + 2
[a__sieve# (x1) ]	=	x1
[a__filter# (x1, x2, x3) ]	=	x1 + 3
[a__zprimes#]	=	0
[mark# (x1) ]	=	2 x1
[0]	=	0
[a__sieve (x1) ]	=	2 x1 + 2
[cons (x1, x2) ]	=	2 x1
[nats (x1) ]	=	2 x1
[a__zprimes]	=	2
[zprimes]	=	2
[a__filter (x1, x2, x3) ]	=	x1 + x2 + x3 + 2
[s (x1) ]	=	2 x1
[a__nats (x1) ]	=	2 x1
[f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

a__sieve#( cons( s( N ) , Y ) )	→	mark#( N )
a__nats#( N )	→	mark#( N )
a__zprimes#	→	a__sieve#( a__nats( s( s( 0 ) ) ) )
a__zprimes#	→	a__nats#( s( s( 0 ) ) )
mark#( nats( X ) )	→	a__nats#( mark( X ) )
mark#( nats( X ) )	→	mark#( X )
mark#( cons( X1 , X2 ) )	→	mark#( X1 )
mark#( s( X ) )	→	mark#( X )

1.1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( nats( X ) )	→	a__nats#( mark( X ) )
  a__nats#( N )	→	mark#( N )
  mark#( nats( X ) )	→	mark#( X )
  mark#( cons( X1 , X2 ) )	→	mark#( X1 )
  mark#( s( X ) )	→	mark#( X )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [a__nats# (x1) ]	=	2 x1 + 1
  [mark (x1) ]	=	x1
  [filter (x1, x2, x3) ]	=	x1 + x2
  [sieve (x1) ]	=	x1
  [mark# (x1) ]	=	2 x1
  [0]	=	0
  [nats (x1) ]	=	x1 + 2
  [cons (x1, x2) ]	=	x1
  [a__sieve (x1) ]	=	x1
  [a__zprimes]	=	2
  [zprimes]	=	2
  [a__filter (x1, x2, x3) ]	=	x1 + x2
  [s (x1) ]	=	2 x1
  [a__nats (x1) ]	=	x1 + 2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( cons( X1 , X2 ) )	→	mark#( X1 )
  mark#( s( X ) )	→	mark#( X )

  1.1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [filter (x1, x2, x3) ]	=	x1
  [mark (x1) ]	=	x1
  [sieve (x1) ]	=	x1
  [mark# (x1) ]	=	2 x1
  [0]	=	0
  [cons (x1, x2) ]	=	x1
  [a__sieve (x1) ]	=	x1
  [nats (x1) ]	=	x1
  [a__zprimes]	=	3
  [a__filter (x1, x2, x3) ]	=	x1
  [zprimes]	=	3
  [s (x1) ]	=	2 x1 + 1
  [a__nats (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( cons( X1 , X2 ) )

  →

  mark#( X1 )

  1.1.1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [filter (x1, x2, x3) ]	=	2 x1 + 2 x2 + 1
  [mark (x1) ]	=	2 x1
  [sieve (x1) ]	=	1
  [mark# (x1) ]	=	2 x1
  [0]	=	1
  [cons (x1, x2) ]	=	x1 + 1
  [a__sieve (x1) ]	=	2
  [nats (x1) ]	=	2 x1 + 2
  [a__zprimes]	=	2
  [a__filter (x1, x2, x3) ]	=	2 x1 + 2 x2 + 2
  [zprimes]	=	2
  [s (x1) ]	=	1
  [a__nats (x1) ]	=	2 x1 + 2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.1.1.1.1.1: P is empty

  All dependency pairs have been removed.