Termination proof

1: switching to dependency pairs

msort#( .( x , y ) )	→	min#( x , y )
msort#( .( x , y ) )	→	msort#( del( min( x , y ) , .( x , y ) ) )
msort#( .( x , y ) )	→	del#( min( x , y ) , .( x , y ) )
msort#( .( x , y ) )	→	min#( x , y )
min#( x , .( y , z ) )	→	min#( x , z )
min#( x , .( y , z ) )	→	min#( y , z )
del#( x , .( y , z ) )	→	del#( x , z )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  msort#( .( x , y ) )

  →

  msort#( del( min( x , y ) , .( x , y ) ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  min#( x , .( y , z ) )	→	min#( y , z )
  min#( x , .( y , z ) )	→	min#( x , z )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [min# (x1, x2) ]	=	2 x1
  [if (x1, x2, x3) ]	=	0
  [msort (x1) ]	=	2 x1
  [<= (x1, x2) ]	=	0
  [= (x1, x2) ]	=	0
  [. (x1, x2) ]	=	x1 + 1
  [nil]	=	0
  [min (x1, x2) ]	=	x1 + 3
  [del (x1, x2) ]	=	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)

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

  →

  del#( x , z )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if (x1, x2, x3) ]	=	0
  [msort (x1) ]	=	2 x1 + 1
  [<= (x1, x2) ]	=	0
  [del# (x1, x2) ]	=	2 x1
  [= (x1, x2) ]	=	0
  [. (x1, x2) ]	=	3 x1 + x2 + 1
  [nil]	=	0
  [min (x1, x2) ]	=	x1
  [del (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

  1.1.3.1: P is empty

  All dependency pairs have been removed.