Termination proof

1: switching to dependency pairs

U11#( tt , N )	→	activate#( N )
U21#( tt , M , N )	→	s#( plus( activate( N ) , activate( M ) ) )
U21#( tt , M , N )	→	plus#( activate( N ) , activate( M ) )
U21#( tt , M , N )	→	activate#( N )
U21#( tt , M , N )	→	activate#( M )
and#( tt , X )	→	activate#( X )
isNat#( n__plus( V1 , V2 ) )	→	and#( isNat( activate( V1 ) ) , n__isNat( activate( V2 ) ) )
isNat#( n__plus( V1 , V2 ) )	→	isNat#( activate( V1 ) )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V1 )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V2 )
isNat#( n__s( V1 ) )	→	isNat#( activate( V1 ) )
isNat#( n__s( V1 ) )	→	activate#( V1 )
plus#( N , 0 )	→	U11#( isNat( N ) , N )
plus#( N , 0 )	→	isNat#( N )
plus#( N , s( M ) )	→	U21#( and( isNat( M ) , n__isNat( N ) ) , M , N )
plus#( N , s( M ) )	→	and#( isNat( M ) , n__isNat( N ) )
plus#( N , s( M ) )	→	isNat#( M )
activate#( n__0 )	→	0#
activate#( n__plus( X1 , X2 ) )	→	plus#( activate( X1 ) , activate( X2 ) )
activate#( n__plus( X1 , X2 ) )	→	activate#( X1 )
activate#( n__plus( X1 , X2 ) )	→	activate#( X2 )
activate#( n__isNat( X ) )	→	isNat#( X )
activate#( n__s( X ) )	→	s#( activate( X ) )
activate#( n__s( X ) )	→	activate#( X )

1.1: dependency graph processor

The dependency pairs are split into 1 component(s).

• The 1st component contains the pair(s)

  activate#( n__plus( X1 , X2 ) )	→	plus#( activate( X1 ) , activate( X2 ) )
  plus#( N , 0 )	→	U11#( isNat( N ) , N )
  U11#( tt , N )	→	activate#( N )
  activate#( n__plus( X1 , X2 ) )	→	activate#( X1 )
  activate#( n__plus( X1 , X2 ) )	→	activate#( X2 )
  activate#( n__isNat( X ) )	→	isNat#( X )
  isNat#( n__plus( V1 , V2 ) )	→	and#( isNat( activate( V1 ) ) , n__isNat( activate( V2 ) ) )
  and#( tt , X )	→	activate#( X )
  activate#( n__s( X ) )	→	activate#( X )
  isNat#( n__plus( V1 , V2 ) )	→	isNat#( activate( V1 ) )
  isNat#( n__plus( V1 , V2 ) )	→	activate#( V1 )
  isNat#( n__plus( V1 , V2 ) )	→	activate#( V2 )
  isNat#( n__s( V1 ) )	→	isNat#( activate( V1 ) )
  isNat#( n__s( V1 ) )	→	activate#( V1 )
  plus#( N , 0 )	→	isNat#( N )
  plus#( N , s( M ) )	→	U21#( and( isNat( M ) , n__isNat( N ) ) , M , N )
  U21#( tt , M , N )	→	plus#( activate( N ) , activate( M ) )
  plus#( N , s( M ) )	→	and#( isNat( M ) , n__isNat( N ) )
  plus#( N , s( M ) )	→	isNat#( M )
  U21#( tt , M , N )	→	activate#( N )
  U21#( tt , M , N )	→	activate#( M )

  1.1.1: reduction pair processor

  Using the following reduction pair

  Linear polynomial interpretation over the naturals

  [n__plus (x1, x2) ]	=	x1 + 2 x2 + 2
  [n__0]	=	1
  [U21 (x1, x2, x3) ]	=	2 x1 + x2 + 3
  [n__s (x1) ]	=	x1 + 1
  [0]	=	1
  [plus# (x1, x2) ]	=	x1 + 2 x2 + 2
  [isNat# (x1) ]	=	x1 + 1
  [activate (x1) ]	=	x1
  [tt]	=	1
  [U21# (x1, x2, x3) ]	=	2 x1 + x2 + 3
  [isNat (x1) ]	=	x1
  [U11 (x1, x2) ]	=	x1 + 2
  [and (x1, x2) ]	=	x1
  [activate# (x1) ]	=	x1 + 2
  [plus (x1, x2) ]	=	x1 + 2 x2 + 2
  [s (x1) ]	=	x1 + 1
  [n__isNat (x1) ]	=	x1
  [U11# (x1, x2) ]	=	x1 + 2
  [and# (x1, x2) ]	=	x1 + x2 + 1
  [f(x1, ..., xn)]	=	x1 + ... + xn + 1	for all other symbols f of arity n

  one remains with the following pair(s).

  U11#( tt , N )	→	activate#( N )
  and#( tt , X )	→	activate#( X )
  isNat#( n__s( V1 ) )	→	activate#( V1 )

  1.1.1.1: dependency graph processor

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