Termination proof

1: switching to dependency pairs

+#( +( x , y ) , z )	→	+#( x , +( y , z ) )
+#( +( x , y ) , z )	→	+#( y , z )
+#( p1 , +( p1 , x ) )	→	+#( p2 , x )
+#( p1 , +( p2 , +( p2 , x ) ) )	→	+#( p5 , x )
+#( p2 , p1 )	→	+#( p1 , p2 )
+#( p2 , +( p1 , x ) )	→	+#( p1 , +( p2 , x ) )
+#( p2 , +( p1 , x ) )	→	+#( p2 , x )
+#( p2 , +( p2 , p2 ) )	→	+#( p1 , p5 )
+#( p2 , +( p2 , +( p2 , x ) ) )	→	+#( p1 , +( p5 , x ) )
+#( p2 , +( p2 , +( p2 , x ) ) )	→	+#( p5 , x )
+#( p5 , p1 )	→	+#( p1 , p5 )
+#( p5 , +( p1 , x ) )	→	+#( p1 , +( p5 , x ) )
+#( p5 , +( p1 , x ) )	→	+#( p5 , x )
+#( p5 , p2 )	→	+#( p2 , p5 )
+#( p5 , +( p2 , x ) )	→	+#( p2 , +( p5 , x ) )
+#( p5 , +( p2 , x ) )	→	+#( p5 , x )
+#( p5 , +( p5 , x ) )	→	+#( p10 , x )
+#( p10 , p1 )	→	+#( p1 , p10 )
+#( p10 , +( p1 , x ) )	→	+#( p1 , +( p10 , x ) )
+#( p10 , +( p1 , x ) )	→	+#( p10 , x )
+#( p10 , p2 )	→	+#( p2 , p10 )
+#( p10 , +( p2 , x ) )	→	+#( p2 , +( p10 , x ) )
+#( p10 , +( p2 , x ) )	→	+#( p10 , x )
+#( p10 , p5 )	→	+#( p5 , p10 )
+#( p10 , +( p5 , x ) )	→	+#( p5 , +( p10 , x ) )
+#( p10 , +( p5 , x ) )	→	+#( p10 , x )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

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

  →

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

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [y]	=	1
  [z]	=	0
  [p1]	=	0
  [p10]	=	0
  [+ (x1, x2) ]	=	2 x1 + x2
  [p2]	=	0
  [+# (x1, x2) ]	=	3 x1
  [p5]	=	0
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  +#( p2 , +( p1 , x ) )	→	+#( p1 , +( p2 , x ) )
  +#( p1 , +( p1 , x ) )	→	+#( p2 , x )
  +#( p2 , +( p1 , x ) )	→	+#( p2 , x )
  +#( p2 , +( p2 , +( p2 , x ) ) )	→	+#( p1 , +( p5 , x ) )
  +#( p1 , +( p2 , +( p2 , x ) ) )	→	+#( p5 , x )
  +#( p5 , +( p1 , x ) )	→	+#( p1 , +( p5 , x ) )
  +#( p5 , +( p1 , x ) )	→	+#( p5 , x )
  +#( p5 , +( p2 , x ) )	→	+#( p2 , +( p5 , x ) )
  +#( p2 , +( p2 , +( p2 , x ) ) )	→	+#( p5 , x )
  +#( p5 , +( p2 , x ) )	→	+#( p5 , x )
  +#( p5 , +( p5 , x ) )	→	+#( p10 , x )
  +#( p10 , +( p1 , x ) )	→	+#( p1 , +( p10 , x ) )
  +#( p10 , +( p1 , x ) )	→	+#( p10 , x )
  +#( p10 , +( p2 , x ) )	→	+#( p2 , +( p10 , x ) )
  +#( p10 , +( p2 , x ) )	→	+#( p10 , x )
  +#( p10 , +( p5 , x ) )	→	+#( p5 , +( p10 , x ) )
  +#( p10 , +( p5 , x ) )	→	+#( p10 , x )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [y]	=	0
  [z]	=	0
  [p1]	=	3
  [p10]	=	0
  [p2]	=	3
  [+ (x1, x2) ]	=	3 x1 + 3 x2
  [+# (x1, x2) ]	=	2 x1 + x2
  [p5]	=	3
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  +#( p2 , +( p1 , x ) )	→	+#( p1 , +( p2 , x ) )
  +#( p5 , +( p1 , x ) )	→	+#( p1 , +( p5 , x ) )
  +#( p5 , +( p2 , x ) )	→	+#( p2 , +( p5 , x ) )

  1.1.2.1: dependency graph processor

  The dependency pairs are split into 0 component(s).