Termination proof

1: switching to dependency pairs

purge#( .( x , y ) )	→	purge#( remove( x , y ) )
purge#( .( x , y ) )	→	remove#( x , y )
remove#( x , .( y , z ) )	→	remove#( x , z )
remove#( x , .( y , z ) )	→	remove#( x , z )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  purge#( .( x , y ) )

  →

  purge#( remove( x , y ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if (x1, x2, x3) ]	=	0
  [purge (x1) ]	=	x1
  [purge# (x1) ]	=	3 x1
  [= (x1, x2) ]	=	0
  [. (x1, x2) ]	=	x1 + 2 x2 + 1
  [nil]	=	0
  [remove (x1, x2) ]	=	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: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  remove#( x , .( y , z ) )

  →

  remove#( x , z )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if (x1, x2, x3) ]	=	3
  [remove# (x1, x2) ]	=	2 x1
  [purge (x1) ]	=	3 x1
  [= (x1, x2) ]	=	0
  [. (x1, x2) ]	=	3 x1 + 2 x2 + 3
  [nil]	=	3
  [remove (x1, x2) ]	=	x1 + x2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.2.1: P is empty

  All dependency pairs have been removed.