Termination proof

1: switching to dependency pairs

merge#( .( x , y ) , .( u , v ) )	→	if#( <( x , u ) , .( x , merge( y , .( u , v ) ) ) , .( u , merge( .( x , y ) , v ) ) )
merge#( .( x , y ) , .( u , v ) )	→	merge#( y , .( u , v ) )
merge#( .( x , y ) , .( u , v ) )	→	merge#( .( x , y ) , v )
++#( .( x , y ) , z )	→	++#( y , z )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  merge#( .( x , y ) , .( u , v ) )	→	merge#( .( x , y ) , v )
  merge#( .( x , y ) , .( u , v ) )	→	merge#( y , .( u , v ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  merge#( .( x , y ) , .( u , v ) )

  →

  merge#( .( x , y ) , v )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

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

  →

  ++#( y , z )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [< (x1, x2) ]	=	0
  [true]	=	0
  [if (x1, x2, x3) ]	=	x1 + 1
  [++ (x1, x2) ]	=	2 x1 + x2 + 3
  [false]	=	0
  [merge (x1, x2) ]	=	2 x1 + x2 + 2
  [. (x1, x2) ]	=	3 x1 + x2 + 3
  [nil]	=	0
  [++# (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.