Termination proof

1: switching to dependency pairs

a__and#( true , X )	→	mark#( X )
a__if#( true , X , Y )	→	mark#( X )
a__if#( false , X , Y )	→	mark#( Y )
a__add#( 0 , X )	→	mark#( X )
mark#( and( X1 , X2 ) )	→	a__and#( mark( X1 ) , X2 )
mark#( and( X1 , X2 ) )	→	mark#( X1 )
mark#( if( X1 , X2 , X3 ) )	→	a__if#( mark( X1 ) , X2 , X3 )
mark#( if( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , X2 )
mark#( add( X1 , X2 ) )	→	mark#( X1 )
mark#( first( X1 , X2 ) )	→	a__first#( mark( X1 ) , mark( X2 ) )
mark#( first( X1 , X2 ) )	→	mark#( X1 )
mark#( first( X1 , X2 ) )	→	mark#( X2 )
mark#( from( X ) )	→	a__from#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  mark#( and( X1 , X2 ) )	→	a__and#( mark( X1 ) , X2 )
  a__and#( true , X )	→	mark#( X )
  mark#( and( X1 , X2 ) )	→	mark#( X1 )
  mark#( if( X1 , X2 , X3 ) )	→	a__if#( mark( X1 ) , X2 , X3 )
  a__if#( true , X , Y )	→	mark#( X )
  mark#( if( X1 , X2 , X3 ) )	→	mark#( X1 )
  mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , X2 )
  a__add#( 0 , X )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	mark#( X1 )
  mark#( first( X1 , X2 ) )	→	mark#( X1 )
  mark#( first( X1 , X2 ) )	→	mark#( X2 )
  a__if#( false , X , Y )	→	mark#( Y )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [from (x1) ]	=	x1
  [mark (x1) ]	=	3 x1 + 2
  [if (x1, x2, x3) ]	=	3 x1 + 2 x2 + 2 x3 + 3
  [first (x1, x2) ]	=	x1 + 2 x2 + 2
  [a__first (x1, x2) ]	=	x1 + 2 x2 + 2
  [a__and# (x1, x2) ]	=	2 x1 + 3 x2
  [mark# (x1) ]	=	3 x1 + 2
  [0]	=	1
  [a__add# (x1, x2) ]	=	2 x1 + 3 x2
  [a__add (x1, x2) ]	=	3 x1 + 3 x2 + 3
  [nil]	=	0
  [cons (x1, x2) ]	=	3 x1 + 1
  [a__and (x1, x2) ]	=	2 x1 + 3 x2 + 3
  [true]	=	2
  [and (x1, x2) ]	=	2 x1 + 2 x2 + 3
  [false]	=	1
  [a__if# (x1, x2, x3) ]	=	3 x1 + 3 x2 + 3 x3 + 2
  [a__if (x1, x2, x3) ]	=	3 x1 + 3 x2 + 3 x3 + 3
  [add (x1, x2) ]	=	3 x1 + 2 x2 + 3
  [a__from (x1) ]	=	3 x1 + 2
  [s (x1) ]	=	0
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  a__add#( 0 , X )

  →

  mark#( X )

  1.1.1.1: dependency graph processor

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