Termination proof

1: switching to dependency pairs

minus#( n__0 , Y )	→	0#
minus#( n__s( X ) , n__s( Y ) )	→	minus#( activate( X ) , activate( Y ) )
minus#( n__s( X ) , n__s( Y ) )	→	activate#( X )
minus#( n__s( X ) , n__s( Y ) )	→	activate#( Y )
geq#( n__s( X ) , n__s( Y ) )	→	geq#( activate( X ) , activate( Y ) )
geq#( n__s( X ) , n__s( Y ) )	→	activate#( X )
geq#( n__s( X ) , n__s( Y ) )	→	activate#( Y )
div#( 0 , n__s( Y ) )	→	0#
div#( s( X ) , n__s( Y ) )	→	if#( geq( X , activate( Y ) ) , n__s( div( minus( X , activate( Y ) ) , n__s( activate( Y ) ) ) ) , n__0 )
div#( s( X ) , n__s( Y ) )	→	geq#( X , activate( Y ) )
div#( s( X ) , n__s( Y ) )	→	activate#( Y )
div#( s( X ) , n__s( Y ) )	→	div#( minus( X , activate( Y ) ) , n__s( activate( Y ) ) )
div#( s( X ) , n__s( Y ) )	→	minus#( X , activate( Y ) )
div#( s( X ) , n__s( Y ) )	→	activate#( Y )
div#( s( X ) , n__s( Y ) )	→	activate#( Y )
if#( true , X , Y )	→	activate#( X )
if#( false , X , Y )	→	activate#( Y )
activate#( n__0 )	→	0#
activate#( n__s( X ) )	→	s#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  div#( s( X ) , n__s( Y ) )

  →

  div#( minus( X , activate( Y ) ) , n__s( activate( Y ) ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	1
  [true]	=	3
  [if (x1, x2, x3) ]	=	3 x1 + 3 x2 + 3 x3
  [n__0]	=	0
  [false]	=	3
  [n__s (x1) ]	=	2
  [s (x1) ]	=	3
  [0]	=	0
  [div# (x1, x2) ]	=	2 x1
  [geq (x1, x2) ]	=	3
  [div (x1, x2) ]	=	3 x1 + 2 x2 + 2
  [activate (x1) ]	=	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)

  minus#( n__s( X ) , n__s( Y ) )

  →

  minus#( activate( X ) , activate( Y ) )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [true]	=	0
  [if (x1, x2, x3) ]	=	2 x1 + 2 x2
  [n__0]	=	0
  [false]	=	0
  [n__s (x1) ]	=	3 x1 + 2
  [0]	=	0
  [s (x1) ]	=	3 x1 + 3
  [minus# (x1, x2) ]	=	3 x1 + x2
  [geq (x1, x2) ]	=	3 x1
  [div (x1, x2) ]	=	3 x1 + 1
  [activate (x1) ]	=	2 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)

  geq#( n__s( X ) , n__s( Y ) )

  →

  geq#( activate( X ) , activate( Y ) )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [true]	=	0
  [if (x1, x2, x3) ]	=	2 x1 + 2 x2
  [n__0]	=	0
  [false]	=	0
  [n__s (x1) ]	=	3 x1 + 2
  [0]	=	0
  [s (x1) ]	=	3 x1 + 3
  [geq (x1, x2) ]	=	3 x1
  [div (x1, x2) ]	=	3 x1 + 1
  [activate (x1) ]	=	2 x1
  [geq# (x1, x2) ]	=	3 x1 + x2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.3.1: P is empty

  All dependency pairs have been removed.