Termination proof

1: switching to dependency pairs

f#( f( X ) )	→	c#( n__f( n__g( n__f( X ) ) ) )
c#( X )	→	d#( activate( X ) )
c#( X )	→	activate#( X )
h#( X )	→	c#( n__d( X ) )
activate#( n__f( X ) )	→	f#( activate( X ) )
activate#( n__f( X ) )	→	activate#( X )
activate#( n__g( X ) )	→	g#( X )
activate#( n__d( X ) )	→	d#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  c#( X )	→	activate#( X )
  activate#( n__f( X ) )	→	f#( activate( X ) )
  f#( f( X ) )	→	c#( n__f( n__g( n__f( X ) ) ) )
  activate#( n__f( X ) )	→	activate#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [h (x1) ]	=	3 x1 + 2
  [c# (x1) ]	=	x1 + 2
  [n__d (x1) ]	=	0
  [activate# (x1) ]	=	x1 + 2
  [f (x1) ]	=	2 x1 + 3
  [d (x1) ]	=	0
  [c (x1) ]	=	x1 + 1
  [n__g (x1) ]	=	0
  [n__f (x1) ]	=	2 x1 + 3
  [f# (x1) ]	=	2 x1 + 2
  [g (x1) ]	=	0
  [activate (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  c#( X )

  →

  activate#( X )

  1.1.1.1: dependency graph processor

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