Termination proof

1: switching to dependency pairs

minus#( X , s( Y ) )	→	pred#( minus( X , Y ) )
minus#( X , s( Y ) )	→	minus#( X , Y )
le#( s( X ) , s( Y ) )	→	le#( X , Y )
gcd#( s( X ) , s( Y ) )	→	if#( le( Y , X ) , s( X ) , s( Y ) )
gcd#( s( X ) , s( Y ) )	→	le#( Y , X )
if#( true , s( X ) , s( Y ) )	→	gcd#( minus( X , Y ) , s( Y ) )
if#( true , s( X ) , s( Y ) )	→	minus#( X , Y )
if#( false , s( X ) , s( Y ) )	→	gcd#( minus( Y , X ) , s( X ) )
if#( false , s( X ) , s( Y ) )	→	minus#( Y , X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  if#( true , s( X ) , s( Y ) )	→	gcd#( minus( X , Y ) , s( Y ) )
  gcd#( s( X ) , s( Y ) )	→	if#( le( Y , X ) , s( X ) , s( Y ) )
  if#( false , s( X ) , s( Y ) )	→	gcd#( minus( Y , X ) , s( X ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [true]	=	0
  [minus (x1, x2) ]	=	x1
  [if (x1, x2, x3) ]	=	x1 + x2
  [gcd# (x1, x2) ]	=	2 x1 + x2
  [false]	=	1
  [if# (x1, x2, x3) ]	=	2 x1 + x2 + x3
  [s (x1) ]	=	2 x1
  [0]	=	1
  [le (x1, x2) ]	=	x1
  [gcd (x1, x2) ]	=	2 x1 + x2
  [pred (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  if#( true , s( X ) , s( Y ) )	→	gcd#( minus( X , Y ) , s( Y ) )
  gcd#( s( X ) , s( Y ) )	→	if#( le( Y , X ) , s( X ) , s( Y ) )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [true]	=	0
  [minus (x1, x2) ]	=	x1
  [if (x1, x2, x3) ]	=	2 x1 + 2 x2 + 2
  [gcd# (x1, x2) ]	=	3 x1 + 3 x2 + 3
  [false]	=	0
  [if# (x1, x2, x3) ]	=	3 x1 + 3 x2
  [s (x1) ]	=	x1 + 3
  [0]	=	1
  [le (x1, x2) ]	=	x1
  [gcd (x1, x2) ]	=	2 x1 + 2 x2 + 2
  [pred (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  minus#( X , s( Y ) )

  →

  minus#( X , Y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	x1
  [true]	=	0
  [if (x1, x2, x3) ]	=	x1 + x2
  [false]	=	0
  [s (x1) ]	=	2 x1 + 2
  [0]	=	0
  [le (x1, x2) ]	=	x1
  [gcd (x1, x2) ]	=	x1 + x2
  [pred (x1) ]	=	x1
  [minus# (x1, x2) ]	=	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)

  le#( s( X ) , s( Y ) )

  →

  le#( X , Y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [minus (x1, x2) ]	=	x1
  [true]	=	2
  [if (x1, x2, x3) ]	=	x1 + x2
  [false]	=	2
  [le# (x1, x2) ]	=	x1
  [s (x1) ]	=	2 x1 + 2
  [0]	=	0
  [le (x1, x2) ]	=	3 x1 + 2
  [gcd (x1, x2) ]	=	x1 + x2
  [pred (x1) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.3.1: P is empty

  All dependency pairs have been removed.