Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( app( nil , YS ) ) → mark#( YS )

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

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

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

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

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

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

active#( from( X ) ) → s#( X )

active#( zWadr( nil , YS ) ) → mark#( nil )

active#( zWadr( XS , nil ) ) → mark#( nil )

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

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

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

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

active#( prefix( L ) ) → prefix#( L )

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

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

mark#( app( X1 , X2 ) ) → mark#( X1 )

mark#( app( X1 , X2 ) ) → mark#( X2 )

mark#( nil ) → active#( nil )

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

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

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

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

mark#( from( X ) ) → from#( mark( X ) )

mark#( from( X ) ) → mark#( X )

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

mark#( s( X ) ) → s#( mark( X ) )

mark#( s( X ) ) → mark#( X )

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

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

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

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

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

mark#( prefix( X ) ) → prefix#( mark( X ) )

mark#( prefix( X ) ) → mark#( X )

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

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

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

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

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 )

from#( mark( X ) ) → from#( X )

from#( active( X ) ) → from#( X )

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

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

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

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

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

prefix#( mark( X ) ) → prefix#( X )

prefix#( active( X ) ) → prefix#( X )

1.1: dependency graph processor

The dependency pairs are split into 7 component(s).

The 1st component contains the pair(s)

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

active#( app( nil , YS ) ) → mark#( YS )

mark#( app( X1 , X2 ) ) → mark#( X1 )

mark#( app( X1 , X2 ) ) → mark#( X2 )

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

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

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

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

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

mark#( from( X ) ) → mark#( X )

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

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

mark#( s( X ) ) → mark#( X )

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

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

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

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

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

mark#( prefix( X ) ) → mark#( X )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 2 x₂

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[s (x₁) ] = 2 x₁ + 3

[prefix (x₁) ] = x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 2 x₁ + 2 x₂

[cons (x₁, x₂) ] = 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#( app( X1 , X2 ) ) → active#( app( mark( X1 ) , mark( X2 ) ) )

active#( app( nil , YS ) ) → mark#( YS )

mark#( app( X1 , X2 ) ) → mark#( X1 )

mark#( app( X1 , X2 ) ) → mark#( X2 )

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

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

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

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

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

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

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

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

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

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

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

mark#( prefix( X ) ) → mark#( X )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[app (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 2

[s (x₁) ] = 1

[prefix (x₁) ] = 1

[nil] = 0

[zWadr (x₁, x₂) ] = 1

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

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

active#( app( nil , YS ) ) → mark#( YS )

mark#( app( X1 , X2 ) ) → mark#( X1 )

mark#( app( X1 , X2 ) ) → mark#( X2 )

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

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

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

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

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

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

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

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

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

mark#( prefix( X ) ) → mark#( X )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[app (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[s (x₁) ] = 0

[prefix (x₁) ] = 1

[nil] = 0

[zWadr (x₁, x₂) ] = 1

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

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

active#( app( nil , YS ) ) → mark#( YS )

mark#( app( X1 , X2 ) ) → mark#( X1 )

mark#( app( X1 , X2 ) ) → mark#( X2 )

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

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

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

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

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

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

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

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

mark#( prefix( X ) ) → mark#( X )

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁ + 2

[app (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁ + 2

[s (x₁) ] = 1

[prefix (x₁) ] = 2 x₁ + 2

[nil] = 0

[zWadr (x₁, x₂) ] = 2 x₁ + x₂ + 3

[cons (x₁, x₂) ] = x₁

[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

[from (x₁) ] = 0

[mark (x₁) ] = 0

[active# (x₁) ] = x₁ + 1

[app (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 2

[s (x₁) ] = 0

[prefix (x₁) ] = 1

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[app (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 2

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 3 x₂ + 2

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[s (x₁) ] = 0

[prefix (x₁) ] = 1

[nil] = 0

[zWadr (x₁, x₂) ] = 3 x₁ + 2 x₂ + 2

[cons (x₁, x₂) ] = 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 )

1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 2

[mark (x₁) ] = x₁

[active (x₁) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + x₂ + 1

[mark# (x₁) ] = 2 x₁

[s (x₁) ] = 2 x₁

[prefix (x₁) ] = x₁ + 2

[nil] = 0

[zWadr (x₁, x₂) ] = 2 x₁ + 2 x₂

[cons (x₁, x₂) ] = 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

1.1.1.1.1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

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

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

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

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

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = 2 x₁ + 3

[active (x₁) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 3

[s (x₁) ] = 2

[app# (x₁, x₂) ] = 2 x₁

[prefix (x₁) ] = 3

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + 3

[cons (x₁, x₂) ] = 0

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

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

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 2

[active (x₁) ] = x₁ + 2

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[app# (x₁, x₂) ] = 2 x₁

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

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

1.1.2.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[app# (x₁, x₂) ] = 2 x₁

[prefix (x₁) ] = 1

[nil] = 0

[zWadr (x₁, x₂) ] = 2

[cons (x₁, x₂) ] = 0

[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 , active( X2 ) ) → app#( X1 , X2 )

1.1.2.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 2

[active (x₁) ] = 2 x₁ + 2

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[app# (x₁, x₂) ] = x₁

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 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

1.1.2.1.1.1.1: 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

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = 2 x₁ + 3

[cons# (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 3

[s (x₁) ] = 2

[prefix (x₁) ] = 3

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + 3

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

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

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

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

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 2

[cons# (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁ + 2

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

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

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

1.1.3.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = x₁ + 1

[cons# (x₁, x₂) ] = 2 x₁

[active (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[prefix (x₁) ] = 1

[nil] = 0

[zWadr (x₁, x₂) ] = 2

[cons (x₁, x₂) ] = 0

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

one remains with the following pair(s).

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

1.1.3.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 2

[cons# (x₁, x₂) ] = x₁

[active (x₁) ] = 2 x₁ + 2

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 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

1.1.3.1.1.1.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

from#( active( X ) ) → from#( X )

from#( mark( X ) ) → from#( X )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[prefix (x₁) ] = 2

[from# (x₁) ] = x₁

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + 1

[cons (x₁, x₂) ] = 1

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

one remains with the following pair(s).

from#( active( X ) ) → from#( X )

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 1

[active (x₁) ] = 2 x₁ + 1

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[from# (x₁) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 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)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[prefix (x₁) ] = 2

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + 1

[cons (x₁, x₂) ] = 1

[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).

s#( active( X ) ) → s#( X )

1.1.5.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 1

[active (x₁) ] = 2 x₁ + 1

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

[s# (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)

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

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

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

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

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = 2 x₁ + 3

[active (x₁) ] = x₁

[app (x₁, x₂) ] = 2 x₁ + 3

[s (x₁) ] = 2

[prefix (x₁) ] = 3

[zWadr# (x₁, x₂) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + 3

[cons (x₁, x₂) ] = 0

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

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

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 2

[active (x₁) ] = x₁ + 2

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[zWadr# (x₁, x₂) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

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

1.1.6.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = x₁ + 3

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[prefix (x₁) ] = 1

[zWadr# (x₁, x₂) ] = 2 x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 2

[cons (x₁, x₂) ] = 0

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

1.1.6.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 2

[active (x₁) ] = 2 x₁ + 2

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[zWadr# (x₁, x₂) ] = x₁

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 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

1.1.6.1.1.1.1: P is empty

All dependency pairs have been removed.

The 7th component contains the pair(s)

prefix#( active( X ) ) → prefix#( X )

prefix#( mark( X ) ) → prefix#( X )

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[app (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[prefix (x₁) ] = 2

[nil] = 0

[zWadr (x₁, x₂) ] = x₁ + 1

[cons (x₁, x₂) ] = 1

[prefix# (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).

prefix#( active( X ) ) → prefix#( X )

1.1.7.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = 1

[active (x₁) ] = 2 x₁ + 1

[app (x₁, x₂) ] = 0

[s (x₁) ] = 0

[prefix (x₁) ] = 0

[nil] = 0

[zWadr (x₁, x₂) ] = 0

[cons (x₁, x₂) ] = 0

[prefix# (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

1.1.7.1.1: P is empty

All dependency pairs have been removed.

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

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

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

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

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