Termination proof

1: switching to dependency pairs

f#( X )	→	if#( X , c , n__f( n__true ) )
if#( false , X , Y )	→	activate#( Y )
activate#( n__f( X ) )	→	f#( activate( X ) )
activate#( n__f( X ) )	→	activate#( X )
activate#( n__true )	→	true#

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  if#( false , X , Y )	→	activate#( Y )
  activate#( n__f( X ) )	→	f#( activate( X ) )
  f#( X )	→	if#( X , c , n__f( n__true ) )
  activate#( n__f( X ) )	→	activate#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [true]	=	0
  [if (x1, x2, x3) ]	=	x1 + 2 x2 + x3
  [n__true]	=	0
  [false]	=	2
  [activate# (x1) ]	=	x1 + 3
  [if# (x1, x2, x3) ]	=	2 x1 + x2
  [c]	=	0
  [f (x1) ]	=	2 x1 + 3
  [n__f (x1) ]	=	2 x1 + 3
  [f# (x1) ]	=	2 x1 + 3
  [activate (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  f#( X )

  →

  if#( X , c , n__f( n__true ) )

  1.1.1.1: dependency graph processor

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