Termination proof

1: switching to dependency pairs

filter#( cons( X , Y ) , 0 , M )	→	activate#( Y )
filter#( cons( X , Y ) , s( N ) , M )	→	activate#( Y )
sieve#( cons( 0 , Y ) )	→	activate#( Y )
sieve#( cons( s( N ) , Y ) )	→	filter#( activate( Y ) , N , N )
sieve#( cons( s( N ) , Y ) )	→	activate#( Y )
zprimes#	→	sieve#( nats( s( s( 0 ) ) ) )
zprimes#	→	nats#( s( s( 0 ) ) )
activate#( n__filter( X1 , X2 , X3 ) )	→	filter#( X1 , X2 , X3 )
activate#( n__sieve( X ) )	→	sieve#( X )
activate#( n__nats( X ) )	→	nats#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  activate#( n__filter( X1 , X2 , X3 ) )	→	filter#( X1 , X2 , X3 )
  filter#( cons( X , Y ) , 0 , M )	→	activate#( Y )
  activate#( n__sieve( X ) )	→	sieve#( X )
  sieve#( cons( 0 , Y ) )	→	activate#( Y )
  sieve#( cons( s( N ) , Y ) )	→	filter#( activate( Y ) , N , N )
  filter#( cons( X , Y ) , s( N ) , M )	→	activate#( Y )
  sieve#( cons( s( N ) , Y ) )	→	activate#( Y )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [filter (x1, x2, x3) ]	=	x1
  [sieve (x1) ]	=	2 x1 + 1
  [n__filter (x1, x2, x3) ]	=	x1
  [n__nats (x1) ]	=	0
  [0]	=	0
  [cons (x1, x2) ]	=	x1
  [nats (x1) ]	=	0
  [activate (x1) ]	=	x1
  [n__sieve (x1) ]	=	2 x1 + 1
  [zprimes]	=	2
  [activate# (x1) ]	=	2 x1
  [sieve# (x1) ]	=	2 x1
  [s (x1) ]	=	0
  [filter# (x1, x2, x3) ]	=	2 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  activate#( n__filter( X1 , X2 , X3 ) )	→	filter#( X1 , X2 , X3 )
  filter#( cons( X , Y ) , 0 , M )	→	activate#( Y )
  sieve#( cons( 0 , Y ) )	→	activate#( Y )
  sieve#( cons( s( N ) , Y ) )	→	filter#( activate( Y ) , N , N )
  filter#( cons( X , Y ) , s( N ) , M )	→	activate#( Y )
  sieve#( cons( s( N ) , Y ) )	→	activate#( Y )

  1.1.1.1: dependency graph processor

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

  • The 1st component contains the pair(s)

    filter#( cons( X , Y ) , 0 , M )	→	activate#( Y )
    activate#( n__filter( X1 , X2 , X3 ) )	→	filter#( X1 , X2 , X3 )
    filter#( cons( X , Y ) , s( N ) , M )	→	activate#( Y )

    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 + x2
    [sieve (x1) ]	=	2
    [n__filter (x1, x2, x3) ]	=	x1 + x2
    [n__nats (x1) ]	=	2 x1
    [0]	=	0
    [cons (x1, x2) ]	=	x1 + x2 + 2
    [nats (x1) ]	=	2 x1 + 2
    [activate (x1) ]	=	2 x1 + 2
    [n__sieve (x1) ]	=	0
    [zprimes]	=	2
    [activate# (x1) ]	=	x1
    [s (x1) ]	=	0
    [filter# (x1, x2, x3) ]	=	x1
    [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

    one remains with the following pair(s).

    activate#( n__filter( X1 , X2 , X3 ) )

    →

    filter#( X1 , X2 , X3 )

    1.1.1.1.1.1: dependency graph processor

    The dependency pairs are split into 0 component(s).