Termination proof

1: switching to dependency pairs

+#( a , b )	→	+#( b , a )
+#( a , +( b , z ) )	→	+#( b , +( a , z ) )
+#( a , +( b , z ) )	→	+#( a , z )
+#( +( x , y ) , z )	→	+#( x , +( y , z ) )
+#( +( x , y ) , z )	→	+#( y , z )
f#( +( x , y ) , z )	→	+#( f( x , z ) , f( y , z ) )
f#( +( x , y ) , z )	→	f#( x , z )
f#( +( x , y ) , z )	→	f#( y , z )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

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

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [b]	=	0
  [a]	=	0
  [+ (x1, x2) ]	=	2 x1 + x2 + 2
  [f (x1, x2) ]	=	2 x1
  [f# (x1, x2) ]	=	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)

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

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [b]	=	0
  [a]	=	0
  [+ (x1, x2) ]	=	2 x1 + x2 + 2
  [f (x1, x2) ]	=	3 x1 + 2
  [+# (x1, x2) ]	=	2 x1 + 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)

  +#( a , +( b , z ) )

  →

  +#( a , z )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [b]	=	2
  [a]	=	2
  [+ (x1, x2) ]	=	2 x1 + x2
  [f (x1, x2) ]	=	x1
  [+# (x1, x2) ]	=	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.