Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( app( cons( X , XS ) , YS ) ) → cons#( X , app( XS , YS ) )

active#( app( cons( X , XS ) , YS ) ) → app#( XS , YS )

active#( from( X ) ) → cons#( X , from( s( X ) ) )

active#( from( X ) ) → from#( s( X ) )

active#( from( X ) ) → s#( X )

active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) ) → cons#( app( Y , cons( X , nil ) ) , zWadr( XS , YS ) )

active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) ) → app#( Y , cons( X , nil ) )

active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) ) → cons#( X , nil )

active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) ) → zWadr#( XS , YS )

active#( prefix( L ) ) → cons#( nil , zWadr( L , prefix( L ) ) )

active#( prefix( L ) ) → zWadr#( L , prefix( L ) )

active#( prefix( L ) ) → prefix#( L )

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

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

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

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

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

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

active#( from( X ) ) → from#( active( X ) )

active#( from( X ) ) → active#( X )

active#( s( X ) ) → s#( active( X ) )

active#( s( X ) ) → active#( X )

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

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

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

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

active#( prefix( X ) ) → prefix#( active( X ) )

active#( prefix( X ) ) → active#( X )

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

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

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

from#( mark( X ) ) → from#( X )

s#( mark( X ) ) → s#( X )

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

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

prefix#( mark( X ) ) → prefix#( X )

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

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

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

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

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

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

proper#( from( X ) ) → from#( proper( X ) )

proper#( from( X ) ) → proper#( X )

proper#( s( X ) ) → s#( proper( X ) )

proper#( s( X ) ) → proper#( X )

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

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

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

proper#( prefix( X ) ) → prefix#( proper( X ) )

proper#( prefix( X ) ) → proper#( X )

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

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

from#( ok( X ) ) → from#( X )

s#( ok( X ) ) → s#( X )

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

prefix#( ok( X ) ) → prefix#( 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 9 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

[from (x₁) ] = 2 x₁ + 1

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 2 x₁ + 2 x₂ + 2

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + x₂ + 1

[s (x₁) ] = 2 x₁

[top# (x₁) ] = x₁

[prefix (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).

top#( ok( X ) ) → top#( active( X ) )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 3 x₁ + 2

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[nil] = 1

[zWadr (x₁, x₂) ] = 3 x₁ + x₂

[cons (x₁, x₂) ] = 3 x₁ + 1

[app (x₁, x₂) ] = x₁

[s (x₁) ] = x₁

[top# (x₁) ] = 2 x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 2 x₁ + 1

[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#( app( X1 , X2 ) ) → active#( X2 )

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

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

active#( from( X ) ) → active#( X )

active#( s( X ) ) → active#( X )

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

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

active#( prefix( X ) ) → active#( X )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 2

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[active (x₁) ] = x₁

[zWadr (x₁, x₂) ] = x₁ + 2 x₂

[nil] = 0

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = 2 x₁

[prefix (x₁) ] = 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).

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

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

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

active#( s( X ) ) → active#( X )

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

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

active#( prefix( X ) ) → active#( X )

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 2 x₁

[zWadr (x₁, x₂) ] = x₁ + x₂ + 2

[nil] = 3

[cons (x₁, x₂) ] = x₁ + x₂

[app (x₁, x₂) ] = x₁ + 3 x₂

[s (x₁) ] = 2 x₁

[prefix (x₁) ] = 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).

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

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

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

active#( s( X ) ) → active#( X )

active#( prefix( X ) ) → active#( X )

1.1.2.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = 0

[active# (x₁) ] = 3 x₁

[active (x₁) ] = 2 x₁

[nil] = 3

[zWadr (x₁, x₂) ] = 2 x₁ + 1

[cons (x₁, x₂) ] = 3 x₁ + 2 x₂

[app (x₁, x₂) ] = 2 x₁ + 3 x₂ + 3

[s (x₁) ] = 2 x₁ + 2

[prefix (x₁) ] = 2 x₁ + 3

[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).

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

1.1.2.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 1

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[active (x₁) ] = x₁

[nil] = 2

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = x₁ + 1

[app (x₁, x₂) ] = x₁ + x₂ + 1

[s (x₁) ] = 1

[prefix (x₁) ] = 1

[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

1.1.2.1.1.1.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

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

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

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

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

proper#( from( X ) ) → proper#( X )

proper#( s( X ) ) → proper#( X )

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

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

proper#( prefix( X ) ) → proper#( X )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 1

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[zWadr (x₁, x₂) ] = 3 x₁ + x₂

[nil] = 0

[cons (x₁, x₂) ] = x₁ + 2 x₂ + 2

[proper# (x₁) ] = 2 x₁

[app (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = 2 x₁

[prefix (x₁) ] = 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).

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

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

proper#( s( X ) ) → proper#( X )

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

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

proper#( prefix( X ) ) → proper#( X )

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 3

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[zWadr (x₁, x₂) ] = 2 x₁ + x₂ + 2

[nil] = 0

[cons (x₁, x₂) ] = 3 x₁ + 2 x₂ + 3

[proper# (x₁) ] = 2 x₁

[app (x₁, x₂) ] = 3 x₁ + 2 x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = 2 x₁ + 3

[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#( app( X1 , X2 ) ) → proper#( X2 )

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

proper#( s( X ) ) → proper#( X )

1.1.3.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 3 x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[nil] = 1

[zWadr (x₁, x₂) ] = 3 x₁

[cons (x₁, x₂) ] = 3 x₁

[proper# (x₁) ] = 2 x₁

[app (x₁, x₂) ] = x₁ + 2 x₂ + 3

[s (x₁) ] = 2 x₁

[prefix (x₁) ] = 2

[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).

proper#( s( X ) ) → proper#( X )

1.1.3.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[nil] = 2

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = 2 x₁

[proper# (x₁) ] = x₁

[app (x₁, x₂) ] = 2

[s (x₁) ] = 2 x₁ + 2

[prefix (x₁) ] = 2 x₁

[ok (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).

none

1.1.3.1.1.1.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

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

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

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

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 3

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[app# (x₁, x₂) ] = x₁ + x₂

[nil] = 0

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁ + 1

[prefix (x₁) ] = x₁

[ok (x₁) ] = x₁ + 1

[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).

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

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

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁ + 3

[active (x₁) ] = x₁ + 3

[app# (x₁, x₂) ] = 3 x₁ + x₂

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + x₂

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.4.1.1: 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

[from (x₁) ] = 2 x₁

[mark (x₁) ] = 0

[active (x₁) ] = 2 x₁

[nil] = 2

[zWadr (x₁, x₂) ] = 2 x₁ + 1

[cons (x₁, x₂) ] = 2 x₁

[cons# (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = 3 x₁ + 3 x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = 2 x₁

[ok (x₁) ] = 2 x₁ + 1

[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).

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

[from (x₁) ] = 2 x₁ + 2

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 2 x₁ + x₂ + 1

[cons (x₁, x₂) ] = 2 x₁ + 3

[cons# (x₁, x₂) ] = 2 x₁

[app (x₁, x₂) ] = 2 x₁ + 2 x₂ + 2

[s (x₁) ] = 2 x₁

[prefix (x₁) ] = x₁ + 3

[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

1.1.5.1.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

from#( ok( X ) ) → from#( X )

from#( mark( X ) ) → from#( X )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[from# (x₁) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 1

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 2 x₁ + 2

[top (x₁) ] = 0

[proper (x₁) ] = 3 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).

from#( mark( X ) ) → from#( X )

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁ + 2

[from# (x₁) ] = x₁

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + x₂

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.6.1.1: P is empty

All dependency pairs have been removed.

The 7th component contains the pair(s)

s#( ok( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = x₁

[s# (x₁) ] = 2 x₁

[app (x₁, x₂) ] = 2 x₁ + 1

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 2 x₁ + 2

[top (x₁) ] = 0

[proper (x₁) ] = 3 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).

s#( mark( X ) ) → s#( X )

1.1.7.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁ + 2

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + x₂

[cons (x₁, x₂) ] = x₁

[s# (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.7.1.1: P is empty

All dependency pairs have been removed.

The 8th component contains the pair(s)

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

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

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

1.1.8: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 3

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[zWadr# (x₁, x₂) ] = x₁ + x₂

[nil] = 0

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = x₁

[s (x₁) ] = 2 x₁ + 1

[prefix (x₁) ] = x₁

[ok (x₁) ] = x₁ + 1

[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).

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

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

1.1.8.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁ + 3

[active (x₁) ] = x₁ + 3

[zWadr# (x₁, x₂) ] = 3 x₁ + x₂

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + x₂

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[proper (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

1.1.8.1.1: P is empty

All dependency pairs have been removed.

The 9th component contains the pair(s)

prefix#( ok( X ) ) → prefix#( X )

prefix#( mark( X ) ) → prefix#( X )

1.1.9: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁

[active (x₁) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = x₁

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 1

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 2 x₁ + 2

[top (x₁) ] = 0

[prefix# (x₁) ] = 2 x₁

[proper (x₁) ] = 3 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).

prefix#( mark( X ) ) → prefix#( X )

1.1.9.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = 2 x₁ + 2

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + x₂

[cons (x₁, x₂) ] = x₁

[app (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = x₁

[prefix (x₁) ] = x₁

[ok (x₁) ] = 0

[top (x₁) ] = 0

[prefix# (x₁) ] = x₁

[proper (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

1.1.9.1.1: P is empty

All dependency pairs have been removed.

active#( app( cons( X , XS ) , YS ) )	→	cons#( X , app( XS , YS ) )
active#( app( cons( X , XS ) , YS ) )	→	app#( XS , YS )
active#( from( X ) )	→	cons#( X , from( s( X ) ) )
active#( from( X ) )	→	from#( s( X ) )
active#( from( X ) )	→	s#( X )
active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) )	→	cons#( app( Y , cons( X , nil ) ) , zWadr( XS , YS ) )
active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) )	→	app#( Y , cons( X , nil ) )
active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) )	→	cons#( X , nil )
active#( zWadr( cons( X , XS ) , cons( Y , YS ) ) )	→	zWadr#( XS , YS )
active#( prefix( L ) )	→	cons#( nil , zWadr( L , prefix( L ) ) )
active#( prefix( L ) )	→	zWadr#( L , prefix( L ) )
active#( prefix( L ) )	→	prefix#( L )
active#( app( X1 , X2 ) )	→	app#( active( X1 ) , X2 )
active#( app( X1 , X2 ) )	→	active#( X1 )
active#( app( X1 , X2 ) )	→	app#( X1 , active( X2 ) )
active#( app( X1 , X2 ) )	→	active#( X2 )
active#( cons( X1 , X2 ) )	→	cons#( active( X1 ) , X2 )
active#( cons( X1 , X2 ) )	→	active#( X1 )
active#( from( X ) )	→	from#( active( X ) )
active#( from( X ) )	→	active#( X )
active#( s( X ) )	→	s#( active( X ) )
active#( s( X ) )	→	active#( X )
active#( zWadr( X1 , X2 ) )	→	zWadr#( active( X1 ) , X2 )
active#( zWadr( X1 , X2 ) )	→	active#( X1 )
active#( zWadr( X1 , X2 ) )	→	zWadr#( X1 , active( X2 ) )
active#( zWadr( X1 , X2 ) )	→	active#( X2 )
active#( prefix( X ) )	→	prefix#( active( X ) )
active#( prefix( X ) )	→	active#( X )
app#( mark( X1 ) , X2 )	→	app#( X1 , X2 )
app#( X1 , mark( X2 ) )	→	app#( X1 , X2 )
cons#( mark( X1 ) , X2 )	→	cons#( X1 , X2 )
from#( mark( X ) )	→	from#( X )
s#( mark( X ) )	→	s#( X )
zWadr#( mark( X1 ) , X2 )	→	zWadr#( X1 , X2 )
zWadr#( X1 , mark( X2 ) )	→	zWadr#( X1 , X2 )
prefix#( mark( X ) )	→	prefix#( X )
proper#( app( X1 , X2 ) )	→	app#( proper( X1 ) , proper( X2 ) )
proper#( app( X1 , X2 ) )	→	proper#( X1 )
proper#( app( X1 , X2 ) )	→	proper#( X2 )
proper#( cons( X1 , X2 ) )	→	cons#( proper( X1 ) , proper( X2 ) )
proper#( cons( X1 , X2 ) )	→	proper#( X1 )
proper#( cons( X1 , X2 ) )	→	proper#( X2 )
proper#( from( X ) )	→	from#( proper( X ) )
proper#( from( X ) )	→	proper#( X )
proper#( s( X ) )	→	s#( proper( X ) )
proper#( s( X ) )	→	proper#( X )
proper#( zWadr( X1 , X2 ) )	→	zWadr#( proper( X1 ) , proper( X2 ) )
proper#( zWadr( X1 , X2 ) )	→	proper#( X1 )
proper#( zWadr( X1 , X2 ) )	→	proper#( X2 )
proper#( prefix( X ) )	→	prefix#( proper( X ) )
proper#( prefix( X ) )	→	proper#( X )
app#( ok( X1 ) , ok( X2 ) )	→	app#( X1 , X2 )
cons#( ok( X1 ) , ok( X2 ) )	→	cons#( X1 , X2 )
from#( ok( X ) )	→	from#( X )
s#( ok( X ) )	→	s#( X )
zWadr#( ok( X1 ) , ok( X2 ) )	→	zWadr#( X1 , X2 )
prefix#( ok( X ) )	→	prefix#( X )
top#( mark( X ) )	→	top#( proper( X ) )
top#( mark( X ) )	→	proper#( X )
top#( ok( X ) )	→	top#( active( X ) )
top#( ok( X ) )	→	active#( X )

[from (x₁) ]	=	2 x₁ + 1
[mark (x₁) ]	=	x₁ + 1
[active (x₁) ]	=	x₁
[nil]	=	0
[zWadr (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 2
[cons (x₁, x₂) ]	=	x₁
[app (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[s (x₁) ]	=	2 x₁
[top# (x₁) ]	=	x₁
[prefix (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

[from (x₁) ]	=	3 x₁ + 2
[mark (x₁) ]	=	0
[active (x₁) ]	=	2 x₁
[nil]	=	1
[zWadr (x₁, x₂) ]	=	3 x₁ + x₂
[cons (x₁, x₂) ]	=	3 x₁ + 1
[app (x₁, x₂) ]	=	x₁
[s (x₁) ]	=	x₁
[top# (x₁) ]	=	2 x₁
[prefix (x₁) ]	=	x₁
[ok (x₁) ]	=	2 x₁ + 1
[top (x₁) ]	=	0
[proper (x₁) ]	=	3 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[from (x₁) ]	=	x₁ + 2
[mark (x₁) ]	=	x₁
[active# (x₁) ]	=	x₁
[active (x₁) ]	=	x₁
[zWadr (x₁, x₂) ]	=	x₁ + 2 x₂
[nil]	=	0
[cons (x₁, x₂) ]	=	x₁
[app (x₁, x₂) ]	=	2 x₁ + x₂
[s (x₁) ]	=	2 x₁
[prefix (x₁) ]	=	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

[from (x₁) ]	=	2 x₁ + 3
[mark (x₁) ]	=	x₁
[active (x₁) ]	=	2 x₁
[app# (x₁, x₂) ]	=	x₁ + x₂
[nil]	=	0
[zWadr (x₁, x₂) ]	=	x₁
[cons (x₁, x₂) ]	=	x₁
[app (x₁, x₂) ]	=	x₁
[s (x₁) ]	=	2 x₁ + 1
[prefix (x₁) ]	=	x₁
[ok (x₁) ]	=	x₁ + 1
[top (x₁) ]	=	0
[proper (x₁) ]	=	2 x₁ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[from (x₁) ]	=	2 x₁ + 2
[mark (x₁) ]	=	x₁ + 1
[active (x₁) ]	=	2 x₁
[nil]	=	0
[zWadr (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[cons (x₁, x₂) ]	=	2 x₁ + 3
[cons# (x₁, x₂) ]	=	2 x₁
[app (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 2
[s (x₁) ]	=	2 x₁
[prefix (x₁) ]	=	x₁ + 3
[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