Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( filter( cons( X , Y ) , 0 , M ) ) → mark#( cons( 0 , filter( Y , M , M ) ) )

active#( filter( cons( X , Y ) , 0 , M ) ) → cons#( 0 , filter( Y , M , M ) )

active#( filter( cons( X , Y ) , 0 , M ) ) → filter#( Y , M , M )

active#( filter( cons( X , Y ) , s( N ) , M ) ) → mark#( cons( X , filter( Y , N , M ) ) )

active#( filter( cons( X , Y ) , s( N ) , M ) ) → cons#( X , filter( Y , N , M ) )

active#( filter( cons( X , Y ) , s( N ) , M ) ) → filter#( Y , N , M )

active#( sieve( cons( 0 , Y ) ) ) → mark#( cons( 0 , sieve( Y ) ) )

active#( sieve( cons( 0 , Y ) ) ) → cons#( 0 , sieve( Y ) )

active#( sieve( cons( 0 , Y ) ) ) → sieve#( Y )

active#( sieve( cons( s( N ) , Y ) ) ) → mark#( cons( s( N ) , sieve( filter( Y , N , N ) ) ) )

active#( sieve( cons( s( N ) , Y ) ) ) → cons#( s( N ) , sieve( filter( Y , N , N ) ) )

active#( sieve( cons( s( N ) , Y ) ) ) → s#( N )

active#( sieve( cons( s( N ) , Y ) ) ) → sieve#( filter( Y , N , N ) )

active#( sieve( cons( s( N ) , Y ) ) ) → filter#( Y , N , N )

active#( nats( N ) ) → mark#( cons( N , nats( s( N ) ) ) )

active#( nats( N ) ) → cons#( N , nats( s( N ) ) )

active#( nats( N ) ) → nats#( s( N ) )

active#( nats( N ) ) → s#( N )

active#( zprimes ) → mark#( sieve( nats( s( s( 0 ) ) ) ) )

active#( zprimes ) → sieve#( nats( s( s( 0 ) ) ) )

active#( zprimes ) → nats#( s( s( 0 ) ) )

active#( zprimes ) → s#( s( 0 ) )

active#( zprimes ) → s#( 0 )

mark#( filter( X1 , X2 , X3 ) ) → active#( filter( mark( X1 ) , mark( X2 ) , mark( X3 ) ) )

mark#( filter( X1 , X2 , X3 ) ) → 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#( cons( X1 , X2 ) ) → active#( cons( mark( X1 ) , X2 ) )

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

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

mark#( 0 ) → active#( 0 )

mark#( s( X ) ) → active#( s( mark( X ) ) )

mark#( s( X ) ) → s#( mark( X ) )

mark#( s( X ) ) → mark#( X )

mark#( sieve( X ) ) → active#( sieve( mark( X ) ) )

mark#( sieve( X ) ) → sieve#( mark( X ) )

mark#( sieve( X ) ) → mark#( X )

mark#( nats( X ) ) → active#( nats( mark( X ) ) )

mark#( nats( X ) ) → nats#( mark( X ) )

mark#( nats( X ) ) → mark#( X )

mark#( zprimes ) → active#( zprimes )

filter#( mark( X1 ) , X2 , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , mark( X2 ) , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , X2 , mark( X3 ) ) → filter#( X1 , X2 , X3 )

filter#( active( X1 ) , X2 , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , active( X2 ) , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , X2 , active( X3 ) ) → filter#( X1 , X2 , X3 )

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

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

cons#( active( X1 ) , X2 ) → cons#( X1 , X2 )

cons#( X1 , active( X2 ) ) → cons#( X1 , X2 )

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

sieve#( mark( X ) ) → sieve#( X )

sieve#( active( X ) ) → sieve#( X )

nats#( mark( X ) ) → nats#( X )

nats#( active( X ) ) → nats#( X )

1.1: dependency graph processor

The dependency pairs are split into 6 component(s).

The 1st component contains the pair(s)

mark#( filter( X1 , X2 , X3 ) ) → active#( filter( mark( X1 ) , mark( X2 ) , mark( X3 ) ) )

active#( filter( cons( X , Y ) , 0 , M ) ) → mark#( cons( 0 , filter( Y , M , M ) ) )

mark#( filter( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( filter( X1 , X2 , X3 ) ) → mark#( X2 )

mark#( filter( X1 , X2 , X3 ) ) → mark#( X3 )

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

active#( filter( cons( X , Y ) , s( N ) , M ) ) → mark#( cons( X , filter( Y , N , M ) ) )

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

mark#( s( X ) ) → active#( s( mark( X ) ) )

active#( sieve( cons( 0 , Y ) ) ) → mark#( cons( 0 , sieve( Y ) ) )

mark#( s( X ) ) → mark#( X )

mark#( sieve( X ) ) → active#( sieve( mark( X ) ) )

active#( sieve( cons( s( N ) , Y ) ) ) → mark#( cons( s( N ) , sieve( filter( Y , N , N ) ) ) )

mark#( sieve( X ) ) → mark#( X )

mark#( nats( X ) ) → active#( nats( mark( X ) ) )

active#( nats( N ) ) → mark#( cons( N , nats( s( N ) ) ) )

mark#( nats( X ) ) → mark#( X )

mark#( zprimes ) → active#( zprimes )

active#( zprimes ) → mark#( sieve( nats( s( s( 0 ) ) ) ) )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[filter (x₁, x₂, x₃) ] = 3 x₁ + x₂ + x₃ + 3

[mark (x₁) ] = 2 x₁

[sieve (x₁) ] = 2 x₁

[active# (x₁) ] = x₁

[zprimes] = 1

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[0] = 0

[s (x₁) ] = 2 x₁

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = 2 x₁

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

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

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

mark#( s( X ) ) → active#( s( mark( X ) ) )

active#( sieve( cons( 0 , Y ) ) ) → mark#( cons( 0 , sieve( Y ) ) )

mark#( s( X ) ) → mark#( X )

mark#( sieve( X ) ) → active#( sieve( mark( X ) ) )

active#( sieve( cons( s( N ) , Y ) ) ) → mark#( cons( s( N ) , sieve( filter( Y , N , N ) ) ) )

mark#( sieve( X ) ) → mark#( X )

mark#( nats( X ) ) → active#( nats( mark( X ) ) )

active#( nats( N ) ) → mark#( cons( N , nats( s( N ) ) ) )

mark#( nats( X ) ) → mark#( X )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[filter (x₁, x₂, x₃) ] = 0

[sieve (x₁) ] = 1

[active# (x₁) ] = x₁

[zprimes] = 0

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[s (x₁) ] = 0

[0] = 0

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = 1

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

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

active#( sieve( cons( 0 , Y ) ) ) → mark#( cons( 0 , sieve( Y ) ) )

mark#( s( X ) ) → mark#( X )

mark#( sieve( X ) ) → active#( sieve( mark( X ) ) )

active#( sieve( cons( s( N ) , Y ) ) ) → mark#( cons( s( N ) , sieve( filter( Y , N , N ) ) ) )

mark#( sieve( X ) ) → mark#( X )

mark#( nats( X ) ) → active#( nats( mark( X ) ) )

active#( nats( N ) ) → mark#( cons( N , nats( s( N ) ) ) )

mark#( nats( X ) ) → mark#( X )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[filter (x₁, x₂, x₃) ] = 2 x₁

[sieve (x₁) ] = x₁

[active# (x₁) ] = x₁

[zprimes] = 1

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 0

[s (x₁) ] = x₁

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = x₁ + 1

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[filter (x₁, x₂, x₃) ] = 0

[sieve (x₁) ] = 1

[active# (x₁) ] = x₁

[zprimes] = 0

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[s (x₁) ] = 0

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[filter (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[sieve (x₁) ] = 2 x₁ + 3

[active# (x₁) ] = 2 x₁

[zprimes] = 3

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁ + 2

[0] = 0

[s (x₁) ] = x₁

[cons (x₁, x₂) ] = 2 x₁

[nats (x₁) ] = 3 x₁

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 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 )

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[filter (x₁, x₂, x₃) ] = 2 x₁

[mark (x₁) ] = x₁

[sieve (x₁) ] = 2 x₁

[zprimes] = 2

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[s (x₁) ] = x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁ + 1

[nats (x₁) ] = 2 x₁ + 1

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

mark#( s( X ) ) → mark#( X )

1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[filter (x₁, x₂, x₃) ] = 2 x₁ + x₂

[mark (x₁) ] = x₁

[sieve (x₁) ] = 0

[zprimes] = 3

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[s (x₁) ] = x₁ + 1

[0] = 0

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = 1

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.1.1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

filter#( X1 , mark( X2 ) , X3 ) → filter#( X1 , X2 , X3 )

filter#( mark( X1 ) , X2 , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , X2 , mark( X3 ) ) → filter#( X1 , X2 , X3 )

filter#( active( X1 ) , X2 , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , active( X2 ) , X3 ) → filter#( X1 , X2 , X3 )

filter#( X1 , X2 , active( X3 ) ) → filter#( X1 , X2 , X3 )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[filter (x₁, x₂, x₃) ] = 0

[sieve (x₁) ] = 0

[zprimes] = 2

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[s (x₁) ] = 0

[filter# (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + x₃

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 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)

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

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

cons#( active( X1 ) , X2 ) → cons#( X1 , X2 )

cons#( X1 , active( X2 ) ) → cons#( X1 , X2 )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2 x₁ + 1

[filter (x₁, x₂, x₃) ] = 1

[sieve (x₁) ] = 1

[zprimes] = 2

[cons# (x₁, x₂) ] = 3 x₁ + x₂

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[s (x₁) ] = 0

[cons (x₁, x₂) ] = 1

[nats (x₁) ] = 2

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 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.

The 4th component contains the pair(s)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = 0

[sieve (x₁) ] = 0

[zprimes] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[s (x₁) ] = 0

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = 0

[s# (x₁) ] = x₁

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.4.1: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

sieve#( active( X ) ) → sieve#( X )

sieve#( mark( X ) ) → sieve#( X )

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = 0

[sieve (x₁) ] = 0

[zprimes] = 0

[sieve# (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[s (x₁) ] = 0

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.5.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

nats#( active( X ) ) → nats#( X )

nats#( mark( X ) ) → nats#( X )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = 0

[nats# (x₁) ] = x₁

[sieve (x₁) ] = 0

[zprimes] = 0

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[s (x₁) ] = 0

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.6.1: P is empty

All dependency pairs have been removed.

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

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

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

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