Termination proof

1: switching to dependency pairs

+#( x , s( y ) )	→	+#( x , y )
+#( s( x ) , y )	→	+#( x , y )
+#( x , +( y , z ) )	→	+#( +( x , y ) , z )
+#( x , +( y , z ) )	→	+#( x , y )
f#( g( f( x ) ) )	→	f#( h( s( 0 ) , x ) )
f#( g( h( x , y ) ) )	→	f#( h( s( x ) , y ) )
f#( h( x , h( y , z ) ) )	→	f#( h( +( x , y ) , z ) )
f#( h( x , h( y , z ) ) )	→	+#( x , y )

1.1: dependency graph processor

The dependency pairs are split into 2 component(s).

• The 1st component contains the pair(s)

  f#( h( x , h( y , z ) ) )

  →

  f#( h( +( x , y ) , z ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  +#( s( x ) , y )	→	+#( x , y )
  +#( x , s( y ) )	→	+#( x , y )
  +#( x , +( y , z ) )	→	+#( +( x , y ) , z )
  +#( x , +( y , z ) )	→	+#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [+ (x1, x2) ]	=	x1 + x2 + 3
  [f (x1) ]	=	0
  [s (x1) ]	=	x1 + 3
  [0]	=	0
  [+# (x1, x2) ]	=	2 x1
  [g (x1) ]	=	0
  [h (x1, x2) ]	=	0
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  +#( s( x ) , y )

  →

  +#( x , y )

  1.1.2.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [+ (x1, x2) ]	=	x1 + 2 x2 + 2
  [f (x1) ]	=	0
  [s (x1) ]	=	x1 + 1
  [0]	=	0
  [+# (x1, x2) ]	=	x1
  [g (x1) ]	=	0
  [h (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.