Termination proof

1: switching to dependency pairs

min#( s( x ) , s( y ) )	→	min#( x , y )
max#( s( x ) , s( y ) )	→	max#( x , y )
-#( s( x ) , s( y ) )	→	-#( x , y )
gcd#( s( x ) , s( y ) , z )	→	gcd#( -( max( x , y ) , min( x , y ) ) , s( min( x , y ) ) , z )
gcd#( s( x ) , s( y ) , z )	→	-#( max( x , y ) , min( x , y ) )
gcd#( s( x ) , s( y ) , z )	→	max#( x , y )
gcd#( s( x ) , s( y ) , z )	→	min#( x , y )
gcd#( s( x ) , s( y ) , z )	→	min#( x , y )
gcd#( x , s( y ) , s( z ) )	→	gcd#( x , -( max( y , z ) , min( y , z ) ) , s( min( y , z ) ) )
gcd#( x , s( y ) , s( z ) )	→	-#( max( y , z ) , min( y , z ) )
gcd#( x , s( y ) , s( z ) )	→	max#( y , z )
gcd#( x , s( y ) , s( z ) )	→	min#( y , z )
gcd#( x , s( y ) , s( z ) )	→	min#( y , z )
gcd#( s( x ) , y , s( z ) )	→	gcd#( -( max( x , z ) , min( x , z ) ) , y , s( min( x , z ) ) )
gcd#( s( x ) , y , s( z ) )	→	-#( max( x , z ) , min( x , z ) )
gcd#( s( x ) , y , s( z ) )	→	max#( x , z )
gcd#( s( x ) , y , s( z ) )	→	min#( x , z )
gcd#( s( x ) , y , s( z ) )	→	min#( x , z )

1.1: dependency graph processor

The dependency pairs are split into 4 component(s).

• The 1st component contains the pair(s)

  gcd#( x , s( y ) , s( z ) )	→	gcd#( x , -( max( y , z ) , min( y , z ) ) , s( min( y , z ) ) )
  gcd#( s( x ) , s( y ) , z )	→	gcd#( -( max( x , y ) , min( x , y ) ) , s( min( x , y ) ) , z )
  gcd#( s( x ) , y , s( z ) )	→	gcd#( -( max( x , z ) , min( x , z ) ) , y , s( min( x , z ) ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max (x1, x2) ]	=	x1 + x2
  [- (x1, x2) ]	=	x1
  [s (x1) ]	=	3 x1 + 1
  [0]	=	0
  [gcd (x1, x2, x3) ]	=	3 x1 + 2 x2 + x3
  [gcd# (x1, x2, x3) ]	=	3 x1 + 2 x2 + x3
  [min (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

  1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  min#( s( x ) , s( y ) )

  →

  min#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [min# (x1, x2) ]	=	x1
  [max (x1, x2) ]	=	x1 + x2
  [- (x1, x2) ]	=	x1
  [s (x1) ]	=	3 x1 + 1
  [0]	=	0
  [gcd (x1, x2, x3) ]	=	3 x1 + 2 x2 + x3
  [min (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)

  max#( s( x ) , s( y ) )

  →

  max#( x , y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max (x1, x2) ]	=	x1 + x2
  [max# (x1, x2) ]	=	x1
  [- (x1, x2) ]	=	x1
  [s (x1) ]	=	3 x1 + 1
  [0]	=	0
  [gcd (x1, x2, x3) ]	=	3 x1 + 2 x2 + x3
  [min (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

  1.1.3.1: P is empty

  All dependency pairs have been removed.

• The 4th component contains the pair(s)

  -#( s( x ) , s( y ) )

  →

  -#( x , y )

  1.1.4: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max (x1, x2) ]	=	x1 + x2
  [- (x1, x2) ]	=	x1
  [s (x1) ]	=	3 x1 + 1
  [0]	=	0
  [gcd (x1, x2, x3) ]	=	3 x1 + 2 x2 + x3
  [min (x1, x2) ]	=	x1
  [-# (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

  1.1.4.1: P is empty

  All dependency pairs have been removed.