Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( and( tt , X ) ) → mark#( X )

active#( plus( N , 0 ) ) → mark#( N )

active#( plus( N , s( M ) ) ) → mark#( s( plus( N , M ) ) )

active#( plus( N , s( M ) ) ) → s#( plus( N , M ) )

active#( plus( N , s( M ) ) ) → plus#( N , M )

mark#( and( X1 , X2 ) ) → active#( and( mark( X1 ) , X2 ) )

mark#( and( X1 , X2 ) ) → and#( mark( X1 ) , X2 )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( tt ) → active#( tt )

mark#( plus( X1 , X2 ) ) → active#( plus( mark( X1 ) , mark( X2 ) ) )

mark#( plus( X1 , X2 ) ) → plus#( mark( X1 ) , mark( X2 ) )

mark#( plus( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → mark#( X2 )

mark#( 0 ) → active#( 0 )

mark#( s( X ) ) → active#( s( mark( X ) ) )

mark#( s( X ) ) → s#( mark( X ) )

mark#( s( X ) ) → mark#( X )

and#( mark( X1 ) , X2 ) → and#( X1 , X2 )

and#( X1 , mark( X2 ) ) → and#( X1 , X2 )

and#( active( X1 ) , X2 ) → and#( X1 , X2 )

and#( X1 , active( X2 ) ) → and#( X1 , X2 )

plus#( mark( X1 ) , X2 ) → plus#( X1 , X2 )

plus#( X1 , mark( X2 ) ) → plus#( X1 , X2 )

plus#( active( X1 ) , X2 ) → plus#( X1 , X2 )

plus#( X1 , active( X2 ) ) → plus#( X1 , X2 )

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

1.1: dependency graph processor

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

The 1st component contains the pair(s)

mark#( and( X1 , X2 ) ) → active#( and( mark( X1 ) , X2 ) )

active#( and( tt , X ) ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → active#( plus( mark( X1 ) , mark( X2 ) ) )

active#( plus( N , 0 ) ) → mark#( N )

mark#( plus( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → mark#( X2 )

mark#( s( X ) ) → active#( s( mark( X ) ) )

active#( plus( N , s( M ) ) ) → mark#( s( plus( N , M ) ) )

mark#( s( X ) ) → mark#( X )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = 1

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[plus (x₁, x₂) ] = 1

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[s (x₁) ] = 0

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

mark#( and( X1 , X2 ) ) → active#( and( mark( X1 ) , X2 ) )

active#( and( tt , X ) ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → active#( plus( mark( X1 ) , mark( X2 ) ) )

active#( plus( N , 0 ) ) → mark#( N )

mark#( plus( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → mark#( X2 )

active#( plus( N , s( M ) ) ) → mark#( s( plus( N , M ) ) )

mark#( s( X ) ) → mark#( X )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = 2 x₁ + x₂

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[plus (x₁, x₂) ] = 2 x₁ + x₂

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 2

[s (x₁) ] = x₁

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

mark#( and( X1 , X2 ) ) → active#( and( mark( X1 ) , X2 ) )

active#( and( tt , X ) ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → active#( plus( mark( X1 ) , mark( X2 ) ) )

mark#( plus( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → mark#( X2 )

active#( plus( N , s( M ) ) ) → mark#( s( plus( N , M ) ) )

mark#( s( X ) ) → mark#( X )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = x₁ + x₂

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + 2 x₂

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[s (x₁) ] = x₁ + 2

[0] = 0

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = 1

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[plus (x₁, x₂) ] = 0

[active (x₁) ] = 0

[mark# (x₁) ] = 2

[0] = 0

[s (x₁) ] = 0

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

mark#( and( X1 , X2 ) ) → active#( and( mark( X1 ) , X2 ) )

active#( and( tt , X ) ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → mark#( X1 )

mark#( plus( X1 , X2 ) ) → mark#( X2 )

1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = 2 x₁ + 2 x₂

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[0] = 0

[s (x₁) ] = 0

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

mark#( and( X1 , X2 ) ) → active#( and( mark( X1 ) , X2 ) )

active#( and( tt , X ) ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = x₁ + 3 x₂ + 2

[mark (x₁) ] = 2 x₁

[active# (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[plus (x₁, x₂) ] = 2 x₁ + 3

[mark# (x₁) ] = 2 x₁

[0] = 3

[s (x₁) ] = 0

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

and#( X1 , mark( X2 ) ) → and#( X1 , X2 )

and#( mark( X1 ) , X2 ) → and#( X1 , X2 )

and#( active( X1 ) , X2 ) → and#( X1 , X2 )

and#( X1 , active( X2 ) ) → and#( X1 , X2 )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3 x₁ + 1

[and (x₁, x₂) ] = 3 x₁ + 1

[active (x₁) ] = x₁ + 3

[plus (x₁, x₂) ] = 3 x₁ + 3

[0] = 2

[s (x₁) ] = 1

[and# (x₁, x₂) ] = 3 x₁ + x₂

[tt] = 3

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 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)

plus#( X1 , mark( X2 ) ) → plus#( X1 , X2 )

plus#( mark( X1 ) , X2 ) → plus#( X1 , X2 )

plus#( active( X1 ) , X2 ) → plus#( X1 , X2 )

plus#( X1 , active( X2 ) ) → plus#( X1 , X2 )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3 x₁ + 1

[and (x₁, x₂) ] = 3 x₁ + 1

[active (x₁) ] = x₁ + 3

[plus (x₁, x₂) ] = 3 x₁ + 3

[0] = 2

[s (x₁) ] = 1

[plus# (x₁, x₂) ] = 3 x₁ + x₂

[tt] = 3

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 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)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[and (x₁, x₂) ] = 2 x₁ + 1

[active (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + 1

[0] = 0

[s (x₁) ] = 0

[s# (x₁) ] = x₁

[tt] = 3

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

s#( active( X ) ) → s#( X )

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[and (x₁, x₂) ] = 0

[mark (x₁) ] = 1

[active (x₁) ] = 2 x₁ + 1

[plus (x₁, x₂) ] = 0

[0] = 0

[s (x₁) ] = 0

[s# (x₁) ] = x₁

[tt] = 0

[f(x₁, ..., x_n)] = x₁ + ... + x_n + 1 for all other symbols f of arity n

one remains with the following pair(s).

none

1.1.4.1.1: P is empty

All dependency pairs have been removed.

active#( and( tt , X ) )	→	mark#( X )
active#( plus( N , 0 ) )	→	mark#( N )
active#( plus( N , s( M ) ) )	→	mark#( s( plus( N , M ) ) )
active#( plus( N , s( M ) ) )	→	s#( plus( N , M ) )
active#( plus( N , s( M ) ) )	→	plus#( N , M )
mark#( and( X1 , X2 ) )	→	active#( and( mark( X1 ) , X2 ) )
mark#( and( X1 , X2 ) )	→	and#( mark( X1 ) , X2 )
mark#( and( X1 , X2 ) )	→	mark#( X1 )
mark#( tt )	→	active#( tt )
mark#( plus( X1 , X2 ) )	→	active#( plus( mark( X1 ) , mark( X2 ) ) )
mark#( plus( X1 , X2 ) )	→	plus#( mark( X1 ) , mark( X2 ) )
mark#( plus( X1 , X2 ) )	→	mark#( X1 )
mark#( plus( X1 , X2 ) )	→	mark#( X2 )
mark#( 0 )	→	active#( 0 )
mark#( s( X ) )	→	active#( s( mark( X ) ) )
mark#( s( X ) )	→	s#( mark( X ) )
mark#( s( X ) )	→	mark#( X )
and#( mark( X1 ) , X2 )	→	and#( X1 , X2 )
and#( X1 , mark( X2 ) )	→	and#( X1 , X2 )
and#( active( X1 ) , X2 )	→	and#( X1 , X2 )
and#( X1 , active( X2 ) )	→	and#( X1 , X2 )
plus#( mark( X1 ) , X2 )	→	plus#( X1 , X2 )
plus#( X1 , mark( X2 ) )	→	plus#( X1 , X2 )
plus#( active( X1 ) , X2 )	→	plus#( X1 , X2 )
plus#( X1 , active( X2 ) )	→	plus#( X1 , X2 )
s#( mark( X ) )	→	s#( X )
s#( active( X ) )	→	s#( X )

[and (x₁, x₂) ]	=	1
[mark (x₁) ]	=	0
[active# (x₁) ]	=	x₁
[plus (x₁, x₂) ]	=	1
[active (x₁) ]	=	0
[mark# (x₁) ]	=	1
[0]	=	0
[s (x₁) ]	=	0
[tt]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[and (x₁, x₂) ]	=	2 x₁ + x₂
[mark (x₁) ]	=	x₁
[active# (x₁) ]	=	x₁
[plus (x₁, x₂) ]	=	2 x₁ + x₂
[active (x₁) ]	=	x₁
[mark# (x₁) ]	=	x₁
[0]	=	2
[s (x₁) ]	=	x₁
[tt]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[and (x₁, x₂) ]	=	x₁ + x₂
[mark (x₁) ]	=	x₁
[active# (x₁) ]	=	x₁
[plus (x₁, x₂) ]	=	x₁ + 2 x₂
[active (x₁) ]	=	x₁
[mark# (x₁) ]	=	x₁
[s (x₁) ]	=	x₁ + 2
[0]	=	0
[tt]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[and (x₁, x₂) ]	=	2 x₁ + 2 x₂
[mark (x₁) ]	=	x₁
[active# (x₁) ]	=	2 x₁
[plus (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 1
[active (x₁) ]	=	x₁
[mark# (x₁) ]	=	2 x₁
[0]	=	0
[s (x₁) ]	=	0
[tt]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[and (x₁, x₂) ]	=	x₁ + 3 x₂ + 2
[mark (x₁) ]	=	2 x₁
[active# (x₁) ]	=	x₁ + 1
[active (x₁) ]	=	x₁
[plus (x₁, x₂) ]	=	2 x₁ + 3
[mark# (x₁) ]	=	2 x₁
[0]	=	3
[s (x₁) ]	=	0
[tt]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	3 x₁ + 1
[and (x₁, x₂) ]	=	3 x₁ + 1
[active (x₁) ]	=	x₁ + 3
[plus (x₁, x₂) ]	=	3 x₁ + 3
[0]	=	2
[s (x₁) ]	=	1
[and# (x₁, x₂) ]	=	3 x₁ + x₂
[tt]	=	3
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n