Termination proof

1: switching to dependency pairs

a__terms#( N )	→	a__sqr#( mark( N ) )
a__terms#( N )	→	mark#( N )
a__add#( 0 , X )	→	mark#( X )
a__first#( s( X ) , cons( Y , Z ) )	→	mark#( Y )
mark#( terms( X ) )	→	a__terms#( mark( X ) )
mark#( terms( X ) )	→	mark#( X )
mark#( sqr( X ) )	→	a__sqr#( mark( X ) )
mark#( sqr( X ) )	→	mark#( X )
mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , mark( X2 ) )
mark#( add( X1 , X2 ) )	→	mark#( X1 )
mark#( add( X1 , X2 ) )	→	mark#( X2 )
mark#( dbl( X ) )	→	a__dbl#( mark( X ) )
mark#( dbl( X ) )	→	mark#( X )
mark#( first( X1 , X2 ) )	→	a__first#( mark( X1 ) , mark( X2 ) )
mark#( first( X1 , X2 ) )	→	mark#( X1 )
mark#( first( X1 , X2 ) )	→	mark#( X2 )
mark#( cons( X1 , X2 ) )	→	mark#( X1 )
mark#( recip( X ) )	→	mark#( X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  a__terms#( N )	→	mark#( N )
  mark#( terms( X ) )	→	a__terms#( mark( X ) )
  mark#( terms( X ) )	→	mark#( X )
  mark#( sqr( X ) )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , mark( X2 ) )
  a__add#( 0 , X )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	mark#( X1 )
  mark#( add( X1 , X2 ) )	→	mark#( X2 )
  mark#( dbl( X ) )	→	mark#( X )
  mark#( first( X1 , X2 ) )	→	a__first#( mark( X1 ) , mark( X2 ) )
  a__first#( s( X ) , cons( Y , Z ) )	→	mark#( Y )
  mark#( first( X1 , X2 ) )	→	mark#( X1 )
  mark#( first( X1 , X2 ) )	→	mark#( X2 )
  mark#( cons( X1 , X2 ) )	→	mark#( X1 )
  mark#( recip( X ) )	→	mark#( X )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [mark (x1) ]	=	x1
  [dbl (x1) ]	=	x1 + 3
  [first (x1, x2) ]	=	2 x1 + x2
  [a__first (x1, x2) ]	=	2 x1 + x2
  [mark# (x1) ]	=	x1
  [0]	=	0
  [a__add# (x1, x2) ]	=	x1
  [a__add (x1, x2) ]	=	2 x1 + 2 x2
  [a__terms# (x1) ]	=	x1
  [nil]	=	0
  [cons (x1, x2) ]	=	2 x1 + 1
  [a__dbl (x1) ]	=	x1 + 3
  [a__terms (x1) ]	=	3 x1 + 3
  [sqr (x1) ]	=	x1
  [terms (x1) ]	=	3 x1 + 3
  [recip (x1) ]	=	x1 + 1
  [add (x1, x2) ]	=	2 x1 + 2 x2
  [s (x1) ]	=	0
  [a__first# (x1, x2) ]	=	x1
  [a__sqr (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  a__terms#( N )	→	mark#( N )
  mark#( sqr( X ) )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , mark( X2 ) )
  a__add#( 0 , X )	→	mark#( X )
  mark#( add( X1 , X2 ) )	→	mark#( X1 )
  mark#( add( X1 , X2 ) )	→	mark#( X2 )
  mark#( first( X1 , X2 ) )	→	a__first#( mark( X1 ) , mark( X2 ) )
  mark#( first( X1 , X2 ) )	→	mark#( X1 )
  mark#( first( X1 , X2 ) )	→	mark#( X2 )

  1.1.1.1: dependency graph processor

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

  • The 1st component contains the pair(s)

    mark#( add( X1 , X2 ) )	→	a__add#( mark( X1 ) , mark( X2 ) )
    a__add#( 0 , X )	→	mark#( X )
    mark#( sqr( X ) )	→	mark#( X )
    mark#( add( X1 , X2 ) )	→	mark#( X1 )
    mark#( add( X1 , X2 ) )	→	mark#( X2 )
    mark#( first( X1 , X2 ) )	→	mark#( X1 )
    mark#( first( X1 , X2 ) )	→	mark#( X2 )

    1.1.1.1.1: reduction pair processor

    Using the following reduction pair

    Linear polynomial interpretation over the naturals

    [mark (x1) ]	=	2 x1
    [dbl (x1) ]	=	0
    [first (x1, x2) ]	=	2 x1 + 2 x2 + 1
    [a__first (x1, x2) ]	=	2 x1 + 2 x2 + 2
    [mark# (x1) ]	=	x1
    [0]	=	0
    [a__add# (x1, x2) ]	=	x1
    [a__add (x1, x2) ]	=	2 x1 + 3 x2 + 3
    [nil]	=	1
    [cons (x1, x2) ]	=	x1 + 3
    [a__dbl (x1) ]	=	0
    [a__terms (x1) ]	=	3
    [terms (x1) ]	=	2
    [recip (x1) ]	=	0
    [sqr (x1) ]	=	2 x1 + 2
    [add (x1, x2) ]	=	2 x1 + 3 x2 + 2
    [s (x1) ]	=	0
    [a__sqr (x1) ]	=	2 x1 + 3
    [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 )

    1.1.1.1.1.1: dependency graph processor

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