Termination proof

1: switching to dependency pairs

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 ) )	→	cons#( X , from( s( X ) ) )
active#( from( X ) )	→	from#( s( X ) )
active#( from( X ) )	→	s#( X )
active#( first( X1 , X2 ) )	→	first#( active( X1 ) , X2 )
active#( first( X1 , X2 ) )	→	active#( X1 )
active#( first( X1 , X2 ) )	→	first#( X1 , active( X2 ) )
active#( first( X1 , X2 ) )	→	active#( X2 )
active#( s( X ) )	→	s#( active( X ) )
active#( s( X ) )	→	active#( X )
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 )
first#( mark( X1 ) , X2 )	→	first#( X1 , X2 )
first#( X1 , mark( X2 ) )	→	first#( X1 , X2 )
s#( mark( X ) )	→	s#( X )
cons#( mark( X1 ) , X2 )	→	cons#( X1 , X2 )
from#( mark( X ) )	→	from#( X )
proper#( first( X1 , X2 ) )	→	first#( proper( X1 ) , proper( X2 ) )
proper#( first( X1 , X2 ) )	→	proper#( X1 )
proper#( first( X1 , X2 ) )	→	proper#( X2 )
proper#( s( X ) )	→	s#( proper( X ) )
proper#( s( X ) )	→	proper#( X )
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 )
first#( ok( X1 ) , ok( X2 ) )	→	first#( X1 , X2 )
s#( ok( X ) )	→	s#( X )
cons#( ok( X1 ) , ok( X2 ) )	→	cons#( X1 , X2 )
from#( ok( X ) )	→	from#( 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 7 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
  [first (x1, x2) ]	=	2 x1 + x2 + 1
  [active (x1) ]	=	x1
  [0]	=	0
  [s (x1) ]	=	x1
  [top# (x1) ]	=	x1
  [ok (x1) ]	=	x1
  [nil]	=	0
  [cons (x1, x2) ]	=	x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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) ]	=	2 x1 + 1
  [mark (x1) ]	=	x1
  [first (x1, x2) ]	=	x1
  [active (x1) ]	=	x1
  [0]	=	0
  [s (x1) ]	=	x1
  [top# (x1) ]	=	x1
  [ok (x1) ]	=	x1 + 1
  [nil]	=	0
  [cons (x1, x2) ]	=	x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	2 x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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#( first( X1 , X2 ) )	→	active#( X2 )
  active#( first( X1 , X2 ) )	→	active#( X1 )
  active#( s( X ) )	→	active#( X )
  active#( cons( X1 , X2 ) )	→	active#( X1 )
  active#( from( X ) )	→	active#( X )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

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

  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) ]	=	x1
  [first (x1, x2) ]	=	3 x1
  [active (x1) ]	=	2 x1
  [s (x1) ]	=	2 x1
  [0]	=	0
  [ok (x1) ]	=	x1
  [nil]	=	0
  [cons (x1, x2) ]	=	x1 + 3
  [top (x1) ]	=	0
  [proper (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  active#( s( X ) )

  →

  active#( X )

  1.1.2.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.2.1.1.1: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

  proper#( first( X1 , X2 ) )	→	proper#( X2 )
  proper#( first( X1 , X2 ) )	→	proper#( X1 )
  proper#( s( X ) )	→	proper#( X )
  proper#( cons( X1 , X2 ) )	→	proper#( X1 )
  proper#( cons( X1 , X2 ) )	→	proper#( X2 )
  proper#( from( X ) )	→	proper#( X )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

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

  1.1.3.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2 x1 + 1
  [mark (x1) ]	=	0
  [proper# (x1) ]	=	2 x1
  [first (x1, x2) ]	=	3 x1
  [active (x1) ]	=	x1
  [s (x1) ]	=	2 x1
  [0]	=	0
  [ok (x1) ]	=	0
  [nil]	=	2
  [cons (x1, x2) ]	=	x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	2 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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

  [from (x1) ]	=	1
  [mark (x1) ]	=	0
  [proper# (x1) ]	=	3 x1
  [first (x1, x2) ]	=	2 x1 + 3 x2
  [active (x1) ]	=	2 x1
  [s (x1) ]	=	x1 + 3
  [0]	=	3
  [ok (x1) ]	=	0
  [nil]	=	3
  [cons (x1, x2) ]	=	3 x1 + 2
  [top (x1) ]	=	0
  [proper (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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)

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

  1.1.4: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  first#( X1 , mark( X2 ) )	→	first#( X1 , X2 )
  first#( mark( X1 ) , X2 )	→	first#( 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 + 1
  [first (x1, x2) ]	=	2 x1 + 2 x2 + 3
  [active (x1) ]	=	2 x1 + 2
  [first# (x1, x2) ]	=	2 x1
  [0]	=	0
  [s (x1) ]	=	x1
  [ok (x1) ]	=	0
  [nil]	=	0
  [cons (x1, x2) ]	=	x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	2 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  first#( X1 , mark( X2 ) )

  →

  first#( X1 , X2 )

  1.1.4.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.4.1.1.1: P is empty

  All dependency pairs have been removed.

• The 5th component contains the pair(s)

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

  one remains with the following pair(s).

  s#( ok( X ) )

  →

  s#( X )

  1.1.5.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	x1
  [mark (x1) ]	=	0
  [first (x1, x2) ]	=	x1
  [active (x1) ]	=	x1
  [0]	=	2
  [s (x1) ]	=	x1
  [ok (x1) ]	=	x1 + 1
  [nil]	=	0
  [cons (x1, x2) ]	=	x1
  [s# (x1) ]	=	2 x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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)

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

  1.1.6: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  cons#( mark( X1 ) , X2 )

  →

  cons#( X1 , X2 )

  1.1.6.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2 x1 + 2
  [mark (x1) ]	=	x1 + 1
  [first (x1, x2) ]	=	x1 + 2 x2
  [cons# (x1, x2) ]	=	x1
  [active (x1) ]	=	x1
  [0]	=	3
  [s (x1) ]	=	x1 + 1
  [ok (x1) ]	=	x1
  [nil]	=	1
  [cons (x1, x2) ]	=	x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	2 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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)

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

  1.1.7: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  from#( ok( X ) )

  →

  from#( X )

  1.1.7.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	x1
  [mark (x1) ]	=	0
  [first (x1, x2) ]	=	x1
  [active (x1) ]	=	x1
  [0]	=	2
  [s (x1) ]	=	x1
  [from# (x1) ]	=	2 x1
  [ok (x1) ]	=	x1 + 1
  [nil]	=	0
  [cons (x1, x2) ]	=	x1
  [top (x1) ]	=	0
  [proper (x1) ]	=	x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 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.