Termination proof

1: switching to dependency pairs

minus#( s( x ) , s( y ) )	→	minus#( x , y )
quot#( s( x ) , s( y ) )	→	quot#( minus( x , y ) , s( y ) )
quot#( s( x ) , s( y ) )	→	minus#( x , y )
app#( add( n , x ) , y )	→	app#( x , y )
reverse#( add( n , x ) )	→	app#( reverse( x ) , add( n , nil ) )
reverse#( add( n , x ) )	→	reverse#( x )
shuffle#( add( n , x ) )	→	shuffle#( reverse( x ) )
shuffle#( add( n , x ) )	→	reverse#( x )
concat#( cons( u , v ) , y )	→	concat#( v , y )
less_leaves#( cons( u , v ) , cons( w , z ) )	→	less_leaves#( concat( u , v ) , concat( w , z ) )
less_leaves#( cons( u , v ) , cons( w , z ) )	→	concat#( u , v )
less_leaves#( cons( u , v ) , cons( w , z ) )	→	concat#( w , z )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  quot#( s( x ) , s( y ) )

  →

  quot#( minus( x , y ) , s( y ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	x1
  [concat (x1, x2) ]	=	2 x1 + 3
  [leaf]	=	0
  [quot# (x1, x2) ]	=	2 x1
  [quot (x1, x2) ]	=	2 x1
  [0]	=	2
  [less_leaves (x1, x2) ]	=	3
  [nil]	=	3
  [shuffle (x1) ]	=	3
  [cons (x1, x2) ]	=	0
  [reverse (x1) ]	=	3 x1 + 3
  [true]	=	2
  [false]	=	2
  [app (x1, x2) ]	=	x1
  [add (x1, x2) ]	=	x1
  [s (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

  1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  minus#( s( x ) , s( y ) )

  →

  minus#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	x1
  [concat (x1, x2) ]	=	2 x1 + x2
  [leaf]	=	0
  [quot (x1, x2) ]	=	x1
  [0]	=	0
  [less_leaves (x1, x2) ]	=	1
  [nil]	=	0
  [shuffle (x1) ]	=	0
  [cons (x1, x2) ]	=	3 x1 + 2
  [reverse (x1) ]	=	2
  [true]	=	0
  [false]	=	1
  [app (x1, x2) ]	=	2 x1 + 1
  [add (x1, x2) ]	=	0
  [s (x1) ]	=	2 x1 + 1
  [minus# (x1, x2) ]	=	2 x1 + 2 x2
  [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: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

  shuffle#( add( n , x ) )

  →

  shuffle#( reverse( x ) )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	2 x1 + 2 x2
  [concat (x1, x2) ]	=	2 x1 + 2 x2
  [leaf]	=	0
  [quot (x1, x2) ]	=	0
  [0]	=	0
  [less_leaves (x1, x2) ]	=	2
  [nil]	=	0
  [shuffle (x1) ]	=	2 x1 + 2
  [cons (x1, x2) ]	=	x1
  [reverse (x1) ]	=	x1
  [true]	=	2
  [shuffle# (x1) ]	=	x1
  [false]	=	0
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	3 x1 + x2 + 1
  [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: P is empty

  All dependency pairs have been removed.

• The 4th component contains the pair(s)

  reverse#( add( n , x ) )

  →

  reverse#( x )

  1.1.4: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	x1 + 3 x2
  [concat (x1, x2) ]	=	2 x1 + x2
  [reverse# (x1) ]	=	x1
  [leaf]	=	0
  [quot (x1, x2) ]	=	0
  [0]	=	0
  [less_leaves (x1, x2) ]	=	0
  [nil]	=	0
  [shuffle (x1) ]	=	2 x1
  [cons (x1, x2) ]	=	x1
  [reverse (x1) ]	=	x1
  [true]	=	0
  [false]	=	0
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	3 x1 + x2 + 2
  [s (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.4.1: P is empty

  All dependency pairs have been removed.

• The 5th component contains the pair(s)

  app#( add( n , x ) , y )

  →

  app#( x , y )

  1.1.5: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	2 x1 + 2 x2
  [concat (x1, x2) ]	=	2 x1 + x2
  [leaf]	=	0
  [quot (x1, x2) ]	=	0
  [0]	=	0
  [app# (x1, x2) ]	=	x1
  [less_leaves (x1, x2) ]	=	2
  [nil]	=	0
  [shuffle (x1) ]	=	2 x1
  [cons (x1, x2) ]	=	x1 + x2
  [reverse (x1) ]	=	x1
  [true]	=	1
  [false]	=	0
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	3 x1 + x2 + 2
  [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

  1.1.5.1: P is empty

  All dependency pairs have been removed.

• The 6th component contains the pair(s)

  less_leaves#( cons( u , v ) , cons( w , z ) )

  →

  less_leaves#( concat( u , v ) , concat( w , z ) )

  1.1.6: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [concat (x1, x2) ]	=	2 x1 + x2
  [minus (x1, x2) ]	=	2 x1 + x2
  [leaf]	=	0
  [quot (x1, x2) ]	=	0
  [0]	=	0
  [less_leaves (x1, x2) ]	=	2
  [nil]	=	0
  [cons (x1, x2) ]	=	2 x1 + x2 + 1
  [shuffle (x1) ]	=	1
  [reverse (x1) ]	=	0
  [true]	=	1
  [less_leaves# (x1, x2) ]	=	x1
  [false]	=	1
  [app (x1, x2) ]	=	2 x1
  [add (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

  1.1.6.1: P is empty

  All dependency pairs have been removed.

• The 7th component contains the pair(s)

  concat#( cons( u , v ) , y )

  →

  concat#( v , y )

  1.1.7: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	x1
  [concat (x1, x2) ]	=	2 x1 + x2 + 2
  [leaf]	=	0
  [quot (x1, x2) ]	=	2 x1
  [0]	=	0
  [less_leaves (x1, x2) ]	=	1
  [nil]	=	0
  [cons (x1, x2) ]	=	2 x1 + x2 + 3
  [shuffle (x1) ]	=	1
  [reverse (x1) ]	=	0
  [true]	=	0
  [false]	=	0
  [app (x1, x2) ]	=	2 x1
  [add (x1, x2) ]	=	0
  [concat# (x1, x2) ]	=	x1
  [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

  1.1.7.1: P is empty

  All dependency pairs have been removed.