Termination proof

1: switching to dependency pairs

minus_active#( s( x ) , s( y ) )	→	minus_active#( x , y )
mark#( s( x ) )	→	mark#( x )
mark#( minus( x , y ) )	→	minus_active#( x , y )
mark#( ge( x , y ) )	→	ge_active#( x , y )
ge_active#( s( x ) , s( y ) )	→	ge_active#( x , y )
mark#( div( x , y ) )	→	div_active#( mark( x ) , y )
mark#( div( x , y ) )	→	mark#( x )
mark#( if( x , y , z ) )	→	if_active#( mark( x ) , y , z )
mark#( if( x , y , z ) )	→	mark#( x )
div_active#( s( x ) , s( y ) )	→	if_active#( ge_active( x , y ) , s( div( minus( x , y ) , s( y ) ) ) , 0 )
div_active#( s( x ) , s( y ) )	→	ge_active#( x , y )
if_active#( true , x , y )	→	mark#( x )
if_active#( false , x , y )	→	mark#( y )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( div( x , y ) )	→	div_active#( mark( x ) , y )
  div_active#( s( x ) , s( y ) )	→	if_active#( ge_active( x , y ) , s( div( minus( x , y ) , s( y ) ) ) , 0 )
  if_active#( true , x , y )	→	mark#( x )
  mark#( s( x ) )	→	mark#( x )
  mark#( div( x , y ) )	→	mark#( x )
  mark#( if( x , y , z ) )	→	if_active#( mark( x ) , y , z )
  if_active#( false , x , y )	→	mark#( y )
  mark#( if( x , y , z ) )	→	mark#( x )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [mark (x1) ]	=	x1
  [if (x1, x2, x3) ]	=	x1 + x2 + 2 x3
  [if_active# (x1, x2, x3) ]	=	2 x1 + 2 x2 + 2 x3
  [mark# (x1) ]	=	2 x1 + 3
  [div_active# (x1, x2) ]	=	3 x1 + 3
  [0]	=	0
  [div (x1, x2) ]	=	3 x1
  [minus_active (x1, x2) ]	=	0
  [ge (x1, x2) ]	=	3 x1 + 2
  [ge_active (x1, x2) ]	=	3 x1 + 2
  [true]	=	2
  [false]	=	2
  [s (x1) ]	=	2 x1 + 1
  [div_active (x1, x2) ]	=	3 x1
  [if_active (x1, x2, x3) ]	=	x1 + x2 + 2 x3
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  mark#( div( x , y ) )	→	div_active#( mark( x ) , y )
  div_active#( s( x ) , s( y ) )	→	if_active#( ge_active( x , y ) , s( div( minus( x , y ) , s( y ) ) ) , 0 )
  mark#( div( x , y ) )	→	mark#( x )
  mark#( if( x , y , z ) )	→	mark#( x )

  1.1.1.1: dependency graph processor

  The dependency pairs are split into 1 component(s).

  • The 1st component contains the pair(s)

    mark#( if( x , y , z ) )	→	mark#( x )
    mark#( div( x , y ) )	→	mark#( x )

    1.1.1.1.1: reduction pair processor

    Using the following reduction pair

    Linear polynomial interpretation over the naturals

    [minus (x1, x2) ]	=	2
    [if (x1, x2, x3) ]	=	x1 + 2 x2 + 2 x3 + 3
    [mark (x1) ]	=	3 x1
    [mark# (x1) ]	=	x1
    [0]	=	0
    [div (x1, x2) ]	=	x1 + x2 + 3
    [minus_active (x1, x2) ]	=	3
    [ge (x1, x2) ]	=	0
    [ge_active (x1, x2) ]	=	0
    [true]	=	0
    [false]	=	0
    [s (x1) ]	=	0
    [div_active (x1, x2) ]	=	x1 + 2 x2 + 3
    [if_active (x1, x2, x3) ]	=	x1 + 3 x2 + 3 x3 + 3
    [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

    one remains with the following pair(s).

    none

    1.1.1.1.1.1: P is empty

    All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  minus_active#( s( x ) , s( y ) )

  →

  minus_active#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [mark (x1) ]	=	x1
  [if (x1, x2, x3) ]	=	x1 + 2 x2
  [0]	=	0
  [minus_active# (x1, x2) ]	=	x1
  [div (x1, x2) ]	=	2 x1
  [minus_active (x1, x2) ]	=	0
  [ge (x1, x2) ]	=	x1 + 2
  [ge_active (x1, x2) ]	=	x1 + 2
  [true]	=	2
  [false]	=	1
  [s (x1) ]	=	x1 + 1
  [div_active (x1, x2) ]	=	2 x1
  [if_active (x1, x2, x3) ]	=	x1 + 2 x2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.2.1: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

  ge_active#( s( x ) , s( y ) )

  →

  ge_active#( x , y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [mark (x1) ]	=	x1
  [if (x1, x2, x3) ]	=	x1 + 2 x2
  [0]	=	0
  [div (x1, x2) ]	=	2 x1
  [minus_active (x1, x2) ]	=	0
  [ge (x1, x2) ]	=	x1 + 2
  [ge_active (x1, x2) ]	=	x1 + 2
  [true]	=	2
  [ge_active# (x1, x2) ]	=	x1
  [false]	=	1
  [s (x1) ]	=	x1 + 1
  [div_active (x1, x2) ]	=	2 x1
  [if_active (x1, x2, x3) ]	=	x1 + 2 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.