Termination proof

1: switching to dependency pairs

a__minus#( s( X ) , s( Y ) )	→	a__minus#( X , Y )
a__geq#( s( X ) , s( Y ) )	→	a__geq#( X , Y )
a__div#( s( X ) , s( Y ) )	→	a__if#( a__geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 )
a__div#( s( X ) , s( Y ) )	→	a__geq#( X , Y )
a__if#( true , X , Y )	→	mark#( X )
a__if#( false , X , Y )	→	mark#( Y )
mark#( minus( X1 , X2 ) )	→	a__minus#( X1 , X2 )
mark#( geq( X1 , X2 ) )	→	a__geq#( X1 , X2 )
mark#( div( X1 , X2 ) )	→	a__div#( mark( X1 ) , X2 )
mark#( div( X1 , X2 ) )	→	mark#( X1 )
mark#( if( X1 , X2 , X3 ) )	→	a__if#( mark( X1 ) , X2 , X3 )
mark#( if( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( s( X ) )	→	mark#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( div( X1 , X2 ) )	→	a__div#( mark( X1 ) , X2 )
  a__div#( s( X ) , s( Y ) )	→	a__if#( a__geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 )
  a__if#( true , X , Y )	→	mark#( X )
  mark#( div( X1 , X2 ) )	→	mark#( X1 )
  mark#( if( X1 , X2 , X3 ) )	→	a__if#( mark( X1 ) , X2 , X3 )
  a__if#( false , X , Y )	→	mark#( Y )
  mark#( if( X1 , X2 , X3 ) )	→	mark#( X1 )
  mark#( s( X ) )	→	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) ]	=	2 x1
  [if (x1, x2, x3) ]	=	2 x1 + x2 + x3 + 2
  [a__div (x1, x2) ]	=	3 x1 + 3
  [mark# (x1) ]	=	2 x1
  [0]	=	0
  [a__minus (x1, x2) ]	=	0
  [a__geq (x1, x2) ]	=	0
  [geq (x1, x2) ]	=	0
  [div (x1, x2) ]	=	3 x1 + 2
  [true]	=	0
  [a__if# (x1, x2, x3) ]	=	2 x1 + 2 x2
  [false]	=	0
  [a__if (x1, x2, x3) ]	=	2 x1 + 2 x2 + 2 x3 + 2
  [s (x1) ]	=	x1 + 3
  [a__div# (x1, x2) ]	=	3 x1 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  a__div#( s( X ) , s( Y ) )	→	a__if#( a__geq( X , Y ) , s( div( minus( X , Y ) , s( Y ) ) ) , 0 )
  a__if#( true , X , Y )	→	mark#( X )
  a__if#( false , X , Y )	→	mark#( Y )

  1.1.1.1: dependency graph processor

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

• The 2nd component contains the pair(s)

  a__minus#( s( X ) , s( Y ) )

  →

  a__minus#( X , Y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [if (x1, x2, x3) ]	=	x1 + 2 x2
  [mark (x1) ]	=	x1
  [a__div (x1, x2) ]	=	2 x1
  [0]	=	0
  [a__minus (x1, x2) ]	=	0
  [a__minus# (x1, x2) ]	=	x1
  [a__geq (x1, x2) ]	=	0
  [div (x1, x2) ]	=	2 x1
  [geq (x1, x2) ]	=	0
  [true]	=	0
  [false]	=	0
  [a__if (x1, x2, x3) ]	=	x1 + 2 x2
  [s (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

  1.1.2.1: P is empty

  All dependency pairs have been removed.

• The 3rd component contains the pair(s)

  a__geq#( s( X ) , s( Y ) )

  →

  a__geq#( X , Y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	0
  [if (x1, x2, x3) ]	=	x1 + 2 x2
  [a__geq# (x1, x2) ]	=	x1
  [mark (x1) ]	=	x1
  [a__div (x1, x2) ]	=	2 x1
  [0]	=	0
  [a__minus (x1, x2) ]	=	0
  [a__geq (x1, x2) ]	=	0
  [div (x1, x2) ]	=	2 x1
  [geq (x1, x2) ]	=	0
  [true]	=	0
  [false]	=	0
  [a__if (x1, x2, x3) ]	=	x1 + 2 x2
  [s (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

  1.1.3.1: P is empty

  All dependency pairs have been removed.