Termination proof

1: switching to dependency pairs

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

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

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

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [u]	=	1
  [+ (x1, x2) ]	=	x1 + x2
  [+# (x1, x2) ]	=	x1 + x2
  [* (x1, x2) ]	=	x1 + x2 + 3
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

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

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [u]	=	0
  [+ (x1, x2) ]	=	x1 + x2 + 1
  [+# (x1, x2) ]	=	2 x1 + x2
  [* (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

  1.1.1.1.1: P is empty

  All dependency pairs have been removed.