Termination proof

1: switching to dependency pairs

lt#( s( X ) , s( Y ) )	→	lt#( X , Y )
append#( add( N , X ) , Y )	→	append#( X , Y )
split#( N , add( M , Y ) )	→	f_1#( split( N , Y ) , N , M , Y )
split#( N , add( M , Y ) )	→	split#( N , Y )
f_1#( pair( X , Z ) , N , M , Y )	→	f_2#( lt( N , M ) , N , M , Y , X , Z )
f_1#( pair( X , Z ) , N , M , Y )	→	lt#( N , M )
qsort#( add( N , X ) )	→	f_3#( split( N , X ) , N , X )
qsort#( add( N , X ) )	→	split#( N , X )
f_3#( pair( Y , Z ) , N , X )	→	append#( qsort( Y ) , add( X , qsort( Z ) ) )
f_3#( pair( Y , Z ) , N , X )	→	qsort#( Y )
f_3#( pair( Y , Z ) , N , X )	→	qsort#( Z )

1.1: dependency graph processor

The dependency pairs are split into 4 component(s).

• The 1st component contains the pair(s)

  qsort#( add( N , X ) )	→	f_3#( split( N , X ) , N , X )
  f_3#( pair( Y , Z ) , N , X )	→	qsort#( Y )
  f_3#( pair( Y , Z ) , N , X )	→	qsort#( Z )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [qsort# (x1) ]	=	2 x1
  [lt (x1, x2) ]	=	0
  [f_1 (x1, ..., x4) ]	=	x1 + 1
  [0]	=	0
  [qsort (x1) ]	=	2 x1
  [nil]	=	0
  [pair (x1, x2) ]	=	x1 + x2
  [true]	=	0
  [append (x1, x2) ]	=	x1 + x2
  [f_3# (x1, x2, x3) ]	=	2 x1 + 2
  [false]	=	0
  [add (x1, x2) ]	=	x1 + 1
  [split (x1, x2) ]	=	x1
  [s (x1) ]	=	0
  [f_2 (x1, ..., x6) ]	=	x1 + x2 + 1
  [f_3 (x1, x2, x3) ]	=	2 x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  qsort#( add( N , X ) )

  →

  f_3#( split( N , X ) , N , X )

  1.1.1.1: dependency graph processor

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

• The 2nd component contains the pair(s)

  split#( N , add( M , Y ) )

  →

  split#( N , Y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [lt (x1, x2) ]	=	2
  [f_1 (x1, ..., x4) ]	=	x1 + 2
  [0]	=	0
  [qsort (x1) ]	=	x1
  [nil]	=	0
  [pair (x1, x2) ]	=	x1 + x2
  [true]	=	2
  [append (x1, x2) ]	=	x1 + x2
  [false]	=	2
  [add (x1, x2) ]	=	x1 + 2
  [split (x1, x2) ]	=	x1
  [s (x1) ]	=	0
  [f_2 (x1, ..., x6) ]	=	x1 + x2 + x3
  [f_3 (x1, x2, x3) ]	=	x1 + 2
  [split# (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

  1.1.2.1: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

  lt#( s( X ) , s( Y ) )

  →

  lt#( X , Y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [lt (x1, x2) ]	=	x1
  [f_1 (x1, ..., x4) ]	=	3 x1
  [0]	=	1
  [qsort (x1) ]	=	0
  [nil]	=	0
  [pair (x1, x2) ]	=	2 x1 + x2
  [true]	=	0
  [append (x1, x2) ]	=	x1
  [false]	=	0
  [add (x1, x2) ]	=	0
  [split (x1, x2) ]	=	0
  [s (x1) ]	=	2 x1 + 1
  [f_2 (x1, ..., x6) ]	=	3 x1 + 2 x2
  [lt# (x1, x2) ]	=	x1
  [f_3 (x1, x2, x3) ]	=	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.

• The 4th component contains the pair(s)

  append#( add( N , X ) , Y )

  →

  append#( X , Y )

  1.1.4: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [lt (x1, x2) ]	=	2
  [f_1 (x1, ..., x4) ]	=	x1 + 2
  [0]	=	0
  [qsort (x1) ]	=	x1
  [nil]	=	0
  [pair (x1, x2) ]	=	x1 + x2
  [append# (x1, x2) ]	=	x1
  [true]	=	2
  [append (x1, x2) ]	=	x1 + x2
  [false]	=	2
  [add (x1, x2) ]	=	x1 + 2
  [split (x1, x2) ]	=	x1
  [s (x1) ]	=	0
  [f_2 (x1, ..., x6) ]	=	x1 + x2 + x3
  [f_3 (x1, x2, x3) ]	=	x1 + 2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.4.1: P is empty

  All dependency pairs have been removed.