Termination proof

1: switching to dependency pairs

le#( s( x ) , s( y ) )	→	le#( x , y )
minus#( s( x ) , y )	→	if_minus#( le( s( x ) , y ) , s( x ) , y )
minus#( s( x ) , y )	→	le#( s( x ) , y )
if_minus#( false , s( x ) , y )	→	minus#( x , y )
mod#( s( x ) , s( y ) )	→	if_mod#( le( y , x ) , s( x ) , s( y ) )
mod#( s( x ) , s( y ) )	→	le#( y , x )
if_mod#( true , s( x ) , s( y ) )	→	mod#( minus( x , y ) , s( y ) )
if_mod#( true , s( x ) , s( y ) )	→	minus#( x , y )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  if_mod#( true , s( x ) , s( y ) )	→	mod#( minus( x , y ) , s( y ) )
  mod#( s( x ) , s( y ) )	→	if_mod#( le( y , x ) , s( x ) , s( y ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

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

  one remains with the following pair(s).

  none

  1.1.1.1: P is empty

  All dependency pairs have been removed.

• The 2nd component contains the pair(s)

  minus#( s( x ) , y )	→	if_minus#( le( s( x ) , y ) , s( x ) , y )
  if_minus#( false , s( x ) , y )	→	minus#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [true]	=	0
  [minus (x1, x2) ]	=	x1
  [false]	=	0
  [if_minus (x1, x2, x3) ]	=	x1
  [s (x1) ]	=	2 x1 + 1
  [0]	=	0
  [if_minus# (x1, x2, x3) ]	=	x1
  [le (x1, x2) ]	=	0
  [minus# (x1, x2) ]	=	2 x1
  [mod (x1, x2) ]	=	2 x1 + 1
  [if_mod (x1, x2, x3) ]	=	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

  [true]	=	0
  [minus (x1, x2) ]	=	2 x1
  [false]	=	0
  [if_minus (x1, x2, x3) ]	=	2 x1
  [le# (x1, x2) ]	=	3 x1 + 3 x2
  [s (x1) ]	=	2 x1 + 2
  [0]	=	1
  [le (x1, x2) ]	=	1
  [mod (x1, x2) ]	=	2 x1 + 2
  [if_mod (x1, x2, x3) ]	=	2 x1 + 1
  [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.