Termination proof

1: switching to dependency pairs

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

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  active#( fst( s( X ) , cons( Y , Z ) ) )	→	mark#( cons( Y , fst( X , Z ) ) )
  mark#( fst( X1 , X2 ) )	→	active#( fst( mark( X1 ) , mark( X2 ) ) )
  active#( from( X ) )	→	mark#( cons( X , from( s( X ) ) ) )
  mark#( fst( X1 , X2 ) )	→	mark#( X1 )
  mark#( fst( X1 , X2 ) )	→	mark#( X2 )
  mark#( s( X ) )	→	active#( s( X ) )
  active#( add( 0 , X ) )	→	mark#( X )
  mark#( cons( X1 , X2 ) )	→	active#( cons( mark( X1 ) , X2 ) )
  active#( add( s( X ) , Y ) )	→	mark#( s( add( X , Y ) ) )
  mark#( cons( X1 , X2 ) )	→	mark#( X1 )
  mark#( from( X ) )	→	active#( from( mark( X ) ) )
  active#( len( cons( X , Z ) ) )	→	mark#( s( len( Z ) ) )
  mark#( from( X ) )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	active#( add( mark( X1 ) , mark( X2 ) ) )
  mark#( add( X1 , X2 ) )	→	mark#( X1 )
  mark#( add( X1 , X2 ) )	→	mark#( X2 )
  mark#( len( X ) )	→	active#( len( mark( X ) ) )
  mark#( len( X ) )	→	mark#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2 x1 + 2
  [len (x1) ]	=	x1 + 2
  [fst (x1, x2) ]	=	2 x1 + 3 x2 + 3
  [mark (x1) ]	=	x1
  [active# (x1) ]	=	2 x1
  [active (x1) ]	=	x1
  [add (x1, x2) ]	=	2 x1 + x2 + 3
  [mark# (x1) ]	=	2 x1 + 1
  [s (x1) ]	=	0
  [0]	=	0
  [nil]	=	0
  [cons (x1, x2) ]	=	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: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

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

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

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

  1.1.2.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  s#( mark( X ) )

  →

  s#( X )

  1.1.3.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2
  [len (x1) ]	=	3
  [fst (x1, x2) ]	=	2 x1 + 1
  [mark (x1) ]	=	2 x1 + 1
  [active (x1) ]	=	x1
  [add (x1, x2) ]	=	2 x1 + 2
  [0]	=	1
  [s (x1) ]	=	0
  [nil]	=	0
  [cons (x1, x2) ]	=	0
  [s# (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

  1.1.3.1.1: 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) ]	=	1
  [len (x1) ]	=	3
  [fst (x1, x2) ]	=	x1
  [mark (x1) ]	=	2 x1 + 1
  [cons# (x1, x2) ]	=	3 x1 + 3 x2
  [active (x1) ]	=	x1
  [add (x1, x2) ]	=	x1 + 3 x2 + 3
  [0]	=	1
  [s (x1) ]	=	1
  [nil]	=	0
  [cons (x1, x2) ]	=	0
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

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

  1.1.4.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  from#( mark( X ) )

  →

  from#( X )

  1.1.5.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2
  [len (x1) ]	=	3
  [fst (x1, x2) ]	=	2 x1 + 1
  [mark (x1) ]	=	2 x1 + 1
  [active (x1) ]	=	x1
  [add (x1, x2) ]	=	2 x1 + 2
  [0]	=	1
  [s (x1) ]	=	0
  [from# (x1) ]	=	2 x1
  [nil]	=	0
  [cons (x1, x2) ]	=	0
  [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)

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

  1.1.6: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

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

  1.1.6.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	0
  [len (x1) ]	=	0
  [fst (x1, x2) ]	=	0
  [mark (x1) ]	=	1
  [active (x1) ]	=	2 x1 + 1
  [add (x1, x2) ]	=	0
  [0]	=	0
  [s (x1) ]	=	0
  [add# (x1, x2) ]	=	3 x1 + 2 x2
  [nil]	=	0
  [cons (x1, x2) ]	=	0
  [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)

  len#( active( X ) )	→	len#( X )
  len#( mark( X ) )	→	len#( X )

  1.1.7: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  len#( mark( X ) )

  →

  len#( X )

  1.1.7.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	2
  [len (x1) ]	=	3
  [fst (x1, x2) ]	=	2 x1 + 1
  [mark (x1) ]	=	2 x1 + 1
  [active (x1) ]	=	x1
  [add (x1, x2) ]	=	2 x1 + 2
  [0]	=	1
  [s (x1) ]	=	0
  [len# (x1) ]	=	2 x1
  [nil]	=	0
  [cons (x1, x2) ]	=	0
  [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.