Termination proof

1: switching to dependency pairs

eq#( s( X ) , s( Y ) )	→	eq#( X , Y )
rm#( N , add( M , X ) )	→	ifrm#( eq( N , M ) , N , add( M , X ) )
rm#( N , add( M , X ) )	→	eq#( N , M )
ifrm#( true , N , add( M , X ) )	→	rm#( N , X )
ifrm#( false , N , add( M , X ) )	→	rm#( N , X )
purge#( add( N , X ) )	→	purge#( rm( N , X ) )
purge#( add( N , X ) )	→	rm#( N , X )

1.1: dependency graph processor

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

• The 1st component contains the pair(s)

  purge#( add( N , X ) )

  →

  purge#( rm( N , X ) )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [true]	=	0
  [false]	=	0
  [add (x1, x2) ]	=	x1 + 1
  [rm (x1, x2) ]	=	x1
  [eq (x1, x2) ]	=	0
  [purge (x1) ]	=	2 x1 + 1
  [0]	=	0
  [s (x1) ]	=	0
  [purge# (x1) ]	=	2 x1
  [ifrm (x1, x2, x3) ]	=	x1
  [nil]	=	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)

  rm#( N , add( M , X ) )	→	ifrm#( eq( N , M ) , N , add( M , X ) )
  ifrm#( true , N , add( M , X ) )	→	rm#( N , X )
  ifrm#( 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

  [ifrm# (x1, x2, x3) ]	=	2 x1 + x2
  [true]	=	2
  [false]	=	2
  [add (x1, x2) ]	=	2 x1 + 2 x2 + 1
  [eq (x1, x2) ]	=	x1
  [rm (x1, x2) ]	=	x1
  [purge (x1) ]	=	2 x1
  [0]	=	2
  [s (x1) ]	=	x1 + 2
  [ifrm (x1, x2, x3) ]	=	x1
  [nil]	=	0
  [rm# (x1, x2) ]	=	2 x1 + 2
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  none

  1.1.2.1: P is empty

  All dependency pairs have been removed.

• 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

  [true]	=	0
  [false]	=	0
  [add (x1, x2) ]	=	x1
  [eq (x1, x2) ]	=	0
  [rm (x1, x2) ]	=	0
  [purge (x1) ]	=	x1
  [s (x1) ]	=	2 x1 + 2
  [0]	=	0
  [ifrm (x1, x2, x3) ]	=	0
  [eq# (x1, x2) ]	=	3 x1 + 3 x2
  [nil]	=	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.