Termination proof

1: switching to dependency pairs

f#( x , g( y , z ) )	→	f#( x , y )
++#( x , g( y , z ) )	→	++#( x , y )
mem#( g( x , y ) , z )	→	mem#( x , z )
mem#( x , max( x ) )	→	null#( x )
max#( g( g( g( x , y ) , z ) , u ) )	→	max#( g( g( x , y ) , z ) )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  f#( x , g( y , z ) )

  →

  f#( x , y )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max' (x1, x2) ]	=	0
  [++ (x1, x2) ]	=	2 x1 + 2 x2
  [or (x1, x2) ]	=	0
  [f# (x1, x2) ]	=	2 x1
  [= (x1, x2) ]	=	0
  [nil]	=	3
  [mem (x1, x2) ]	=	2 x1
  [true]	=	0
  [max (x1) ]	=	0
  [u]	=	0
  [g (x1, x2) ]	=	x1 + 3
  [false]	=	0
  [f (x1, x2) ]	=	3 x1
  [null (x1) ]	=	1
  [not (x1) ]	=	0
  [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)

  ++#( x , g( y , z ) )

  →

  ++#( x , y )

  1.1.2: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max' (x1, x2) ]	=	0
  [++ (x1, x2) ]	=	2 x1 + 2 x2
  [or (x1, x2) ]	=	0
  [= (x1, x2) ]	=	0
  [nil]	=	3
  [mem (x1, x2) ]	=	2 x1
  [++# (x1, x2) ]	=	2 x1
  [true]	=	0
  [max (x1) ]	=	0
  [u]	=	0
  [g (x1, x2) ]	=	x1 + 3
  [false]	=	0
  [f (x1, x2) ]	=	3 x1
  [null (x1) ]	=	1
  [not (x1) ]	=	0
  [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)

  mem#( g( x , y ) , z )

  →

  mem#( x , z )

  1.1.3: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max' (x1, x2) ]	=	0
  [++ (x1, x2) ]	=	2 x1 + 2 x2
  [or (x1, x2) ]	=	0
  [= (x1, x2) ]	=	0
  [nil]	=	3
  [mem (x1, x2) ]	=	2 x1
  [true]	=	0
  [max (x1) ]	=	0
  [u]	=	0
  [g (x1, x2) ]	=	x1 + 3
  [false]	=	0
  [f (x1, x2) ]	=	3 x1
  [null (x1) ]	=	1
  [mem# (x1, x2) ]	=	2 x1
  [not (x1) ]	=	0
  [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)

  max#( g( g( g( x , y ) , z ) , u ) )

  →

  max#( g( g( x , y ) , z ) )

  1.1.4: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [max' (x1, x2) ]	=	1
  [++ (x1, x2) ]	=	x1 + x2
  [or (x1, x2) ]	=	x1
  [= (x1, x2) ]	=	1
  [nil]	=	3
  [mem (x1, x2) ]	=	2 x1 + 3 x2 + 2
  [true]	=	0
  [max (x1) ]	=	1
  [g (x1, x2) ]	=	x1 + 2
  [u]	=	0
  [false]	=	1
  [f (x1, x2) ]	=	3 x1 + 3 x2
  [null (x1) ]	=	x1 + 2
  [max# (x1) ]	=	x1
  [not (x1) ]	=	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

  1.1.4.1: P is empty

  All dependency pairs have been removed.