Termination proof

1: switching to dependency pairs

app#( app( plus , app( s , x ) ) , y )	→	app#( s , app( app( plus , x ) , y ) )
app#( app( plus , app( s , x ) ) , y )	→	app#( app( plus , x ) , y )
app#( app( plus , app( s , x ) ) , y )	→	app#( plus , x )
app#( app( app( curry , f ) , x ) , y )	→	app#( app( f , x ) , y )
app#( app( app( curry , f ) , x ) , y )	→	app#( f , x )
add#	→	app#( curry , plus )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  app#( app( app( curry , f ) , x ) , y )	→	app#( app( f , x ) , y )
  app#( app( plus , app( s , x ) ) , y )	→	app#( app( plus , x ) , y )
  app#( app( app( curry , f ) , x ) , y )	→	app#( f , x )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [curry]	=	0
  [app (x1, x2) ]	=	x1 + x2 + 1
  [plus]	=	0
  [0]	=	0
  [app# (x1, x2) ]	=	x1
  [s]	=	2
  [add]	=	2
  [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.