Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( first( 0 , X ) ) → mark#( nil )

active#( first( s( X ) , cons( Y , Z ) ) ) → mark#( cons( Y , first( X , Z ) ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → cons#( Y , first( X , Z ) )

active#( first( s( X ) , cons( Y , Z ) ) ) → first#( X , Z )

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 )

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

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

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

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

mark#( 0 ) → active#( 0 )

mark#( nil ) → active#( nil )

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

mark#( s( X ) ) → s#( mark( X ) )

mark#( s( X ) ) → mark#( X )

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 )

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

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

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

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

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

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 )

1.1: dependency graph processor

The dependency pairs are split into 5 component(s).

The 1st component contains the pair(s)

active#( first( s( X ) , cons( Y , Z ) ) ) → mark#( cons( Y , first( X , Z ) ) )

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

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

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

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

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

mark#( s( X ) ) → mark#( X )

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

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

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

mark#( from( X ) ) → mark#( X )

1.1.1: 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 + 1

[first (x₁, x₂) ] = 2 x₁ + 2 x₂

[active (x₁) ] = x₁

[mark# (x₁) ] = 3 x₁ + 1

[s (x₁) ] = x₁ + 1

[0] = 1

[nil] = 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#( first( X1 , X2 ) ) → active#( first( mark( X1 ) , mark( X2 ) ) )

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

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

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

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

1.1.1.1: dependency graph processor

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

The 1st component contains the pair(s)

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

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

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

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 1

[mark (x₁) ] = x₁

[first (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 1

[s (x₁) ] = 0

[nil] = 1

[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: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2 x₁ + 2

[mark (x₁) ] = x₁

[first (x₁, x₂) ] = x₁ + x₂

[active (x₁) ] = x₁

[mark# (x₁) ] = 3 x₁

[0] = 0

[s (x₁) ] = 0

[nil] = 0

[cons (x₁, x₂) ] = 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

1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

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

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

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

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

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2

[mark (x₁) ] = 2 x₁ + 1

[first (x₁, x₂) ] = 3

[active (x₁) ] = x₁ + 3

[first# (x₁, x₂) ] = 3 x₁ + x₂

[0] = 3

[s (x₁) ] = 2

[nil] = 2

[cons (x₁, x₂) ] = 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

1.1.2.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = x₁ + 1

[first (x₁, x₂) ] = 2

[active (x₁) ] = x₁ + 1

[0] = 0

[s (x₁) ] = 0

[nil] = 1

[cons (x₁, x₂) ] = 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

1.1.3.1: P is empty

All dependency pairs have been removed.

The 4th 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 )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 2

[mark (x₁) ] = 2 x₁ + 1

[first (x₁, x₂) ] = 3

[cons# (x₁, x₂) ] = 3 x₁ + x₂

[active (x₁) ] = x₁ + 3

[0] = 3

[s (x₁) ] = 2

[nil] = 2

[cons (x₁, x₂) ] = 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

1.1.4.1: P is empty

All dependency pairs have been removed.

The 5th component contains the pair(s)

from#( active( X ) ) → from#( X )

from#( mark( X ) ) → from#( X )

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[from (x₁) ] = 0

[mark (x₁) ] = x₁ + 1

[first (x₁, x₂) ] = 2

[active (x₁) ] = x₁ + 1

[0] = 0

[s (x₁) ] = 0

[from# (x₁) ] = x₁

[nil] = 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).

none

1.1.5.1: P is empty

All dependency pairs have been removed.

active#( first( 0 , X ) )	→	mark#( nil )
active#( first( s( X ) , cons( Y , Z ) ) )	→	mark#( cons( Y , first( X , Z ) ) )
active#( first( s( X ) , cons( Y , Z ) ) )	→	cons#( Y , first( X , Z ) )
active#( first( s( X ) , cons( Y , Z ) ) )	→	first#( X , Z )
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 )
mark#( first( X1 , X2 ) )	→	active#( first( mark( X1 ) , mark( X2 ) ) )
mark#( first( X1 , X2 ) )	→	first#( mark( X1 ) , mark( X2 ) )
mark#( first( X1 , X2 ) )	→	mark#( X1 )
mark#( first( X1 , X2 ) )	→	mark#( X2 )
mark#( 0 )	→	active#( 0 )
mark#( nil )	→	active#( nil )
mark#( s( X ) )	→	active#( s( mark( X ) ) )
mark#( s( X ) )	→	s#( mark( X ) )
mark#( s( X ) )	→	mark#( X )
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 )
first#( mark( X1 ) , X2 )	→	first#( X1 , X2 )
first#( X1 , mark( X2 ) )	→	first#( X1 , X2 )
first#( active( X1 ) , X2 )	→	first#( X1 , X2 )
first#( X1 , active( X2 ) )	→	first#( X1 , X2 )
s#( mark( X ) )	→	s#( X )
s#( active( X ) )	→	s#( X )
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 )

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

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