Termination proof

1: switching to dependency pairs

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 )

1.1: dependency graph processor

The dependency pairs are split into 3 component(s).

• The 1st component contains the pair(s)

  shuffle#( add( n , x ) )

  →

  shuffle#( reverse( x ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [reverse (x1) ]	=	x1
  [shuffle# (x1) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	x1 + 2
  [nil]	=	0
  [shuffle (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

  1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  reverse#( add( n , x ) )

  →

  reverse#( x )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [reverse (x1) ]	=	x1
  [reverse# (x1) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	x1 + 1
  [nil]	=	0
  [shuffle (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: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

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

  →

  app#( x , y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [reverse (x1) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	x1 + 1
  [app# (x1, x2) ]	=	x1
  [nil]	=	0
  [shuffle (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: P is empty

  All dependency pairs have been removed.