Termination proof

1: switching to dependency pairs

f#( g( x ) , g( y ) )	→	f#( p( f( g( x ) , s( y ) ) ) , g( s( p( x ) ) ) )
f#( g( x ) , g( y ) )	→	p#( f( g( x ) , s( y ) ) )
f#( g( x ) , g( y ) )	→	f#( g( x ) , s( y ) )
f#( g( x ) , g( y ) )	→	g#( x )
f#( g( x ) , g( y ) )	→	g#( s( p( x ) ) )
f#( g( x ) , g( y ) )	→	p#( x )
p#( 0 )	→	g#( 0 )
g#( s( p( x ) ) )	→	p#( x )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  f#( g( x ) , g( y ) )

  →

  f#( p( f( g( x ) , s( y ) ) ) , g( s( p( x ) ) ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [f (x1, x2) ]	=	0
  [s (x1) ]	=	x1
  [0]	=	3
  [f# (x1, x2) ]	=	2 x1
  [g (x1) ]	=	2 x1 + 2
  [p (x1) ]	=	3 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.