Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

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

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

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

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

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 ) ) → cons#( N , nats( s( N ) ) )

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

active#( nats( N ) ) → s#( N )

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

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

active#( zprimes ) → s#( s( 0 ) )

active#( zprimes ) → s#( 0 )

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

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

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

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

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

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

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

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

active#( s( X ) ) → s#( active( X ) )

active#( s( X ) ) → active#( X )

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

active#( sieve( X ) ) → active#( X )

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

active#( nats( X ) ) → active#( X )

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 )

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

s#( mark( X ) ) → s#( X )

sieve#( mark( X ) ) → sieve#( X )

nats#( mark( X ) ) → nats#( X )

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

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

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

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

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

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

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

proper#( s( X ) ) → s#( proper( X ) )

proper#( s( X ) ) → proper#( X )

proper#( sieve( X ) ) → sieve#( proper( X ) )

proper#( sieve( X ) ) → proper#( X )

proper#( nats( X ) ) → nats#( proper( X ) )

proper#( nats( X ) ) → proper#( X )

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

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

s#( ok( X ) ) → s#( X )

sieve#( ok( X ) ) → sieve#( X )

nats#( ok( X ) ) → nats#( X )

top#( mark( X ) ) → top#( proper( X ) )

top#( mark( X ) ) → proper#( X )

top#( ok( X ) ) → top#( active( X ) )

top#( ok( X ) ) → active#( X )

1.1: dependency graph processor

The dependency pairs are split into 8 component(s).

The 1st component contains the pair(s)

top#( ok( X ) ) → top#( active( X ) )

top#( mark( X ) ) → top#( proper( X ) )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

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

[sieve (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁

[nats (x₁) ] = 2 x₁ + 1

[zprimes] = 3

[s (x₁) ] = x₁

[top# (x₁) ] = 2 x₁

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

top#( ok( X ) ) → top#( active( X ) )

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) ] = 0

[sieve (x₁) ] = 2 x₁ + 2

[active (x₁) ] = x₁

[0] = 3

[cons (x₁, x₂) ] = 2 x₁ + 2 x₂ + 3

[nats (x₁) ] = 2 x₁

[zprimes] = 3

[s (x₁) ] = x₁

[top# (x₁) ] = 3 x₁

[ok (x₁) ] = 2 x₁ + 3

[top (x₁) ] = 0

[proper (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).

none

1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

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

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

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

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

active#( s( X ) ) → active#( X )

active#( sieve( X ) ) → active#( X )

active#( nats( X ) ) → active#( X )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

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

[mark (x₁) ] = 0

[sieve (x₁) ] = 2 x₁

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 2 x₁

[0] = 2

[cons (x₁, x₂) ] = 2 x₁ + 1

[nats (x₁) ] = 2 x₁ + 2

[zprimes] = 0

[s (x₁) ] = x₁ + 2

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

active#( sieve( X ) ) → active#( X )

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[filter (x₁, x₂, x₃) ] = 2 x₁ + 2

[mark (x₁) ] = 0

[sieve (x₁) ] = x₁ + 1

[active# (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[0] = 2

[cons (x₁, x₂) ] = 0

[nats (x₁) ] = x₁

[zprimes] = 0

[s (x₁) ] = 2

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

none

1.1.2.1.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

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

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

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

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

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

proper#( s( X ) ) → proper#( X )

proper#( sieve( X ) ) → proper#( X )

proper#( nats( X ) ) → proper#( X )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

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

[mark (x₁) ] = 3

[sieve (x₁) ] = 2 x₁ + 2

[active (x₁) ] = 2 x₁

[0] = 2

[cons (x₁, x₂) ] = 2 x₁ + 2 x₂

[nats (x₁) ] = 2 x₁ + 2

[zprimes] = 2

[proper# (x₁) ] = 2 x₁

[s (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

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

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

proper#( s( X ) ) → proper#( X )

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

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

[mark (x₁) ] = 0

[sieve (x₁) ] = 1

[active (x₁) ] = 2 x₁

[0] = 3

[cons (x₁, x₂) ] = 2 x₁ + 2 x₂ + 3

[nats (x₁) ] = 0

[zprimes] = 3

[proper# (x₁) ] = 2 x₁

[s (x₁) ] = 2 x₁

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

proper#( s( X ) ) → proper#( X )

1.1.3.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[filter (x₁, x₂, x₃) ] = 3 x₁

[mark (x₁) ] = 0

[sieve (x₁) ] = 3

[active (x₁) ] = x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁ + 3

[nats (x₁) ] = 0

[zprimes] = 0

[proper# (x₁) ] = x₁

[s (x₁) ] = x₁ + 1

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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

1.1.3.1.1.1: P is empty

All dependency pairs have been removed.

The 4th 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#( ok( X1 ) , ok( X2 ) , ok( X3 ) ) → filter#( X1 , X2 , X3 )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[filter (x₁, x₂, x₃) ] = x₁

[sieve (x₁) ] = x₁

[active (x₁) ] = x₁

[0] = 0

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = x₁

[zprimes] = 0

[s (x₁) ] = x₁

[filter# (x₁, x₂, x₃) ] = x₁ + 3 x₂

[ok (x₁) ] = 2 x₁ + 3

[top (x₁) ] = 0

[proper (x₁) ] = 3

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

one remains with the following 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 )

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

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

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁ + 2

[nats (x₁) ] = x₁ + 2

[zprimes] = 2

[s (x₁) ] = 2 x₁

[filter# (x₁, x₂, x₃) ] = 2 x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

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

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

1.1.4.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

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

[sieve (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁

[nats (x₁) ] = x₁ + 2

[zprimes] = 3

[s (x₁) ] = x₁

[filter# (x₁, x₂, x₃) ] = 2 x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

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

1.1.4.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 2

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

[sieve (x₁) ] = x₁ + 2

[active (x₁) ] = 2 x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁

[nats (x₁) ] = 2 x₁ + 2

[zprimes] = 3

[s (x₁) ] = x₁

[filter# (x₁, x₂, x₃) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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).

none

1.1.4.1.1.1.1: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

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

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

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[filter (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[sieve (x₁) ] = x₁ + 2

[active (x₁) ] = 2 x₁

[0] = 2

[cons (x₁, x₂) ] = 2 x₁ + 2 x₂

[nats (x₁) ] = 2 x₁

[zprimes] = 2

[cons# (x₁, x₂) ] = 2 x₁

[s (x₁) ] = 2 x₁ + 2

[ok (x₁) ] = x₁ + 2

[top (x₁) ] = 0

[proper (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).

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

1.1.5.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = x₁ + x₂ + x₃

[sieve (x₁) ] = x₁ + 3

[active (x₁) ] = x₁ + 1

[0] = 0

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = x₁

[zprimes] = 3

[cons# (x₁, x₂) ] = x₁

[s (x₁) ] = x₁

[ok (x₁) ] = x₁

[top (x₁) ] = 0

[proper (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).

none

1.1.5.1.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

s#( ok( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[filter (x₁, x₂, x₃) ] = x₁

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 2

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = x₁

[s# (x₁) ] = 2 x₁

[zprimes] = 3

[s (x₁) ] = x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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).

s#( mark( X ) ) → s#( X )

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = x₁ + x₂ + x₃

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 0

[cons (x₁, x₂) ] = x₁ + 1

[nats (x₁) ] = x₁

[s# (x₁) ] = x₁

[zprimes] = 1

[s (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.6.1.1: P is empty

All dependency pairs have been removed.

The 7th component contains the pair(s)

sieve#( ok( X ) ) → sieve#( X )

sieve#( mark( X ) ) → sieve#( X )

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[filter (x₁, x₂, x₃) ] = x₁

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 2

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = x₁

[zprimes] = 3

[sieve# (x₁) ] = 2 x₁

[s (x₁) ] = x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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).

sieve#( mark( X ) ) → sieve#( X )

1.1.7.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = x₁ + x₂ + x₃

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 0

[cons (x₁, x₂) ] = x₁ + 1

[nats (x₁) ] = x₁

[zprimes] = 1

[sieve# (x₁) ] = x₁

[s (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.7.1.1: P is empty

All dependency pairs have been removed.

The 8th component contains the pair(s)

nats#( ok( X ) ) → nats#( X )

nats#( mark( X ) ) → nats#( X )

1.1.8: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[filter (x₁, x₂, x₃) ] = x₁

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 2

[cons (x₁, x₂) ] = x₁

[nats (x₁) ] = x₁

[nats# (x₁) ] = 2 x₁

[zprimes] = 3

[s (x₁) ] = x₁

[ok (x₁) ] = x₁ + 1

[top (x₁) ] = 0

[proper (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).

nats#( mark( X ) ) → nats#( X )

1.1.8.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[filter (x₁, x₂, x₃) ] = x₁ + x₂ + x₃

[sieve (x₁) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[0] = 0

[cons (x₁, x₂) ] = x₁ + 1

[nats (x₁) ] = x₁

[nats# (x₁) ] = x₁

[zprimes] = 1

[s (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.8.1.1: P is empty

All dependency pairs have been removed.

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

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

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

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

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