Termination proof

1: switching to dependency pairs

eq#( s( x ) , s( y ) )	→	eq#( x , y )
le#( s( x ) , s( y ) )	→	le#( x , y )
app#( add( n , x ) , y )	→	app#( x , y )
min#( add( n , add( m , x ) ) )	→	if_min#( le( n , m ) , add( n , add( m , x ) ) )
min#( add( n , add( m , x ) ) )	→	le#( n , m )
if_min#( true , add( n , add( m , x ) ) )	→	min#( add( n , x ) )
if_min#( false , add( n , add( m , x ) ) )	→	min#( add( m , x ) )
rm#( n , add( m , x ) )	→	if_rm#( eq( n , m ) , n , add( m , x ) )
rm#( n , add( m , x ) )	→	eq#( n , m )
if_rm#( true , n , add( m , x ) )	→	rm#( n , x )
if_rm#( false , n , add( m , x ) )	→	rm#( n , x )
minsort#( add( n , x ) , y )	→	if_minsort#( eq( n , min( add( n , x ) ) ) , add( n , x ) , y )
minsort#( add( n , x ) , y )	→	eq#( n , min( add( n , x ) ) )
minsort#( add( n , x ) , y )	→	min#( add( n , x ) )
if_minsort#( true , add( n , x ) , y )	→	minsort#( app( rm( n , x ) , y ) , nil )
if_minsort#( true , add( n , x ) , y )	→	app#( rm( n , x ) , y )
if_minsort#( true , add( n , x ) , y )	→	rm#( n , x )
if_minsort#( false , add( n , x ) , y )	→	minsort#( x , add( n , y ) )

1.1: dependency graph processor

The dependency pairs are split into 6 component(s).

• The 1st component contains the pair(s)

  if_minsort#( true , add( n , x ) , y )	→	minsort#( app( rm( n , x ) , y ) , nil )
  minsort#( add( n , x ) , y )	→	if_minsort#( eq( n , min( add( n , x ) ) ) , add( n , x ) , y )
  if_minsort#( false , add( n , x ) , y )	→	minsort#( x , add( n , y ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	2 x1 + 2 x2
  [rm (x1, x2) ]	=	x1 + x2
  [eq (x1, x2) ]	=	0
  [0]	=	3
  [nil]	=	0
  [if_min (x1, x2) ]	=	x1
  [minsort# (x1, x2) ]	=	x1 + x2
  [minsort (x1, x2) ]	=	2 x1 + 2 x2
  [true]	=	0
  [if_minsort# (x1, x2, x3) ]	=	x1 + x2
  [false]	=	0
  [if_rm (x1, x2, x3) ]	=	x1 + x2
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	2 x1 + x2 + 2
  [s (x1) ]	=	2 x1 + 3
  [min (x1) ]	=	x1
  [le (x1, x2) ]	=	3 x1 + 3 x2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  minsort#( add( n , x ) , y )	→	if_minsort#( eq( n , min( add( n , x ) ) ) , add( n , x ) , y )
  if_minsort#( false , add( n , x ) , y )	→	minsort#( x , add( n , y ) )

  1.1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	2 x1 + 2 x2
  [rm (x1, x2) ]	=	x1
  [eq (x1, x2) ]	=	0
  [0]	=	0
  [nil]	=	0
  [if_min (x1, x2) ]	=	2 x1
  [minsort# (x1, x2) ]	=	x1 + 2
  [minsort (x1, x2) ]	=	2 x1 + 2 x2
  [true]	=	0
  [if_minsort# (x1, x2, x3) ]	=	x1 + 1
  [false]	=	0
  [if_rm (x1, x2, x3) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	x1 + x2 + 1
  [s (x1) ]	=	2 x1
  [min (x1) ]	=	2 x1
  [le (x1, x2) ]	=	3 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  if_minsort#( false , add( n , x ) , y )

  →

  minsort#( x , add( n , y ) )

  1.1.1.1.1: dependency graph processor

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

• The 2nd component contains the pair(s)

  rm#( n , add( m , x ) )	→	if_rm#( eq( n , m ) , n , add( m , x ) )
  if_rm#( true , n , add( m , x ) )	→	rm#( n , x )
  if_rm#( false , n , add( m , x ) )	→	rm#( n , x )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	2 x1 + 2 x2
  [rm (x1, x2) ]	=	x1 + x2
  [eq (x1, x2) ]	=	0
  [0]	=	0
  [nil]	=	0
  [if_min (x1, x2) ]	=	2 x1 + 1
  [minsort (x1, x2) ]	=	2 x1 + 2 x2
  [true]	=	0
  [false]	=	0
  [if_rm (x1, x2, x3) ]	=	x1 + x2
  [app (x1, x2) ]	=	x1 + x2 + 1
  [add (x1, x2) ]	=	2 x1 + x2 + 2
  [min (x1) ]	=	2 x1 + 2
  [s (x1) ]	=	0
  [le (x1, x2) ]	=	0
  [if_rm# (x1, x2, x3) ]	=	x1
  [rm# (x1, x2) ]	=	x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  rm#( n , add( m , x ) )

  →

  if_rm#( eq( n , m ) , n , add( m , x ) )

  1.1.2.1: dependency graph processor

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

• The 3rd component contains the pair(s)

  eq#( s( x ) , s( y ) )

  →

  eq#( x , y )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	x1 + x2
  [eq (x1, x2) ]	=	2 x1
  [rm (x1, x2) ]	=	x1
  [0]	=	2
  [eq# (x1, x2) ]	=	x1
  [nil]	=	0
  [if_min (x1, x2) ]	=	2 x1 + 1
  [minsort (x1, x2) ]	=	x1 + x2
  [true]	=	3
  [false]	=	0
  [if_rm (x1, x2, x3) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	2 x1 + x2
  [min (x1) ]	=	2 x1 + 1
  [s (x1) ]	=	2 x1 + 1
  [le (x1, x2) ]	=	3 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

  1.1.3.1: P is empty

  All dependency pairs have been removed.

• The 4th component contains the pair(s)

  min#( add( n , add( m , x ) ) )	→	if_min#( le( n , m ) , add( n , add( m , x ) ) )
  if_min#( true , add( n , add( m , x ) ) )	→	min#( add( n , x ) )
  if_min#( false , add( n , add( m , x ) ) )	→	min#( add( m , x ) )

  1.1.4: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	x1 + x2
  [rm (x1, x2) ]	=	x1
  [eq (x1, x2) ]	=	0
  [0]	=	0
  [nil]	=	0
  [if_min (x1, x2) ]	=	x1
  [minsort (x1, x2) ]	=	x1 + x2
  [min# (x1) ]	=	2 x1 + 3
  [true]	=	0
  [false]	=	0
  [if_rm (x1, x2, x3) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	2 x1 + x2 + 2
  [min (x1) ]	=	x1 + 2
  [s (x1) ]	=	0
  [le (x1, x2) ]	=	0
  [if_min# (x1, x2) ]	=	2 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.

• The 5th component contains the pair(s)

  le#( s( x ) , s( y ) )

  →

  le#( x , y )

  1.1.5: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	x1 + x2
  [le# (x1, x2) ]	=	x1
  [eq (x1, x2) ]	=	2 x1
  [rm (x1, x2) ]	=	x1
  [0]	=	2
  [nil]	=	0
  [if_min (x1, x2) ]	=	2 x1 + 1
  [minsort (x1, x2) ]	=	x1 + x2
  [true]	=	3
  [false]	=	0
  [if_rm (x1, x2, x3) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	2 x1 + x2
  [min (x1) ]	=	2 x1 + 1
  [s (x1) ]	=	2 x1 + 1
  [le (x1, x2) ]	=	3 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

  1.1.5.1: P is empty

  All dependency pairs have been removed.

• The 6th component contains the pair(s)

  app#( add( n , x ) , y )

  →

  app#( x , y )

  1.1.6: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [if_minsort (x1, x2, x3) ]	=	2 x1 + 2 x2
  [eq (x1, x2) ]	=	2
  [rm (x1, x2) ]	=	x1
  [0]	=	1
  [app# (x1, x2) ]	=	2 x1
  [nil]	=	0
  [if_min (x1, x2) ]	=	2 x1
  [minsort (x1, x2) ]	=	2 x1 + 2 x2
  [true]	=	0
  [false]	=	2
  [if_rm (x1, x2, x3) ]	=	x1
  [app (x1, x2) ]	=	x1 + x2
  [add (x1, x2) ]	=	x1 + x2 + 1
  [min (x1) ]	=	2 x1 + 1
  [s (x1) ]	=	3 x1
  [le (x1, x2) ]	=	3 x1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.6.1: P is empty

  All dependency pairs have been removed.