Termination proof

1: switching to dependency pairs

qsort#( .( x , y ) )	→	qsort#( lowers( x , y ) )
qsort#( .( x , y ) )	→	lowers#( x , y )
qsort#( .( x , y ) )	→	qsort#( greaters( x , y ) )
qsort#( .( x , y ) )	→	greaters#( x , y )
lowers#( x , .( y , z ) )	→	lowers#( x , z )
lowers#( x , .( y , z ) )	→	lowers#( x , z )
greaters#( x , .( y , z ) )	→	greaters#( x , z )
greaters#( x , .( y , z ) )	→	greaters#( x , z )

1.1: dependency graph processor

The dependency pairs are split into 3 component(s).

• The 1st component contains the pair(s)

  qsort#( .( x , y ) )	→	qsort#( greaters( x , y ) )
  qsort#( .( x , y ) )	→	qsort#( lowers( x , y ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [lowers (x1, x2) ]	=	2 x1 + 1
  [++ (x1, x2) ]	=	0
  [if (x1, x2, x3) ]	=	x1 + x2
  [qsort# (x1) ]	=	2 x1
  [greaters (x1, x2) ]	=	x1 + 1
  [<= (x1, x2) ]	=	x1
  [qsort (x1) ]	=	2 x1
  [. (x1, x2) ]	=	2 x1 + 2 x2 + 2
  [nil]	=	3
  [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)

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

  →

  lowers#( x , z )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.2.1: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

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

  →

  greaters#( x , z )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.3.1: P is empty

  All dependency pairs have been removed.