Termination proof

1: switching to dependency pairs

+#( x , minus( y ) )	→	minus#( +( minus( x ) , y ) )
+#( x , minus( y ) )	→	+#( minus( x ) , y )
+#( x , minus( y ) )	→	minus#( x )
+#( x , +( y , z ) )	→	+#( +( x , y ) , z )
+#( x , +( y , z ) )	→	+#( x , y )
+#( minus( +( x , 1 ) ) , 1 )	→	minus#( x )

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

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.1.1: P is empty

  All dependency pairs have been removed.