Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

a__U11#( tt , N ) → mark#( N )

a__U21#( tt , M , N ) → a__plus#( mark( N ) , mark( M ) )

a__U21#( tt , M , N ) → mark#( N )

a__U21#( tt , M , N ) → mark#( M )

a__and#( tt , X ) → mark#( X )

a__isNat#( plus( V1 , V2 ) ) → a__and#( a__isNat( V1 ) , isNat( V2 ) )

a__isNat#( plus( V1 , V2 ) ) → a__isNat#( V1 )

a__isNat#( s( V1 ) ) → a__isNat#( V1 )

a__plus#( N , 0 ) → a__U11#( a__isNat( N ) , N )

a__plus#( N , 0 ) → a__isNat#( N )

a__plus#( N , s( M ) ) → a__U21#( a__and( a__isNat( M ) , isNat( N ) ) , M , N )

a__plus#( N , s( M ) ) → a__and#( a__isNat( M ) , isNat( N ) )

a__plus#( N , s( M ) ) → a__isNat#( M )

mark#( U11( X1 , X2 ) ) → a__U11#( mark( X1 ) , X2 )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U21( X1 , X2 , X3 ) ) → a__U21#( mark( X1 ) , X2 , X3 )

mark#( U21( X1 , X2 , X3 ) ) → mark#( X1 )

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

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

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

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

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isNat( X ) ) → a__isNat#( X )

mark#( s( X ) ) → mark#( X )

1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U11 (x₁, x₂) ] = 2 x₁ + x₂ + 3

[mark (x₁) ] = x₁

[a__U21# (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[U21 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + x₃ + 3

[a__and# (x₁, x₂) ] = 2 x₁ + 1

[mark# (x₁) ] = x₁ + 1

[a__plus (x₁, x₂) ] = x₁ + x₂ + 3

[0] = 3

[tt] = 0

[a__isNat# (x₁) ] = 1

[a__and (x₁, x₂) ] = x₁ + 2 x₂

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 2 x₁ + x₂ + 3

[and (x₁, x₂) ] = x₁ + 2 x₂

[a__U21 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + x₃ + 3

[plus (x₁, x₂) ] = x₁ + x₂ + 3

[a__U11# (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = x₁

[a__isNat (x₁) ] = 0

[a__plus# (x₁, x₂) ] = x₁ + x₂ + 1

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

one remains with the following pair(s).

a__U21#( tt , M , N ) → a__plus#( mark( N ) , mark( M ) )

a__U21#( tt , M , N ) → mark#( N )

a__U21#( tt , M , N ) → mark#( M )

a__and#( tt , X ) → mark#( X )

a__isNat#( plus( V1 , V2 ) ) → a__and#( a__isNat( V1 ) , isNat( V2 ) )

a__isNat#( plus( V1 , V2 ) ) → a__isNat#( V1 )

a__isNat#( s( V1 ) ) → a__isNat#( V1 )

a__plus#( N , s( M ) ) → a__U21#( a__and( a__isNat( M ) , isNat( N ) ) , M , N )

a__plus#( N , s( M ) ) → a__and#( a__isNat( M ) , isNat( N ) )

a__plus#( N , s( M ) ) → a__isNat#( M )

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

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isNat( X ) ) → a__isNat#( X )

mark#( s( X ) ) → mark#( X )

1.1.1: dependency graph processor

The dependency pairs are split into 2 component(s).

The 1st component contains the pair(s)

a__plus#( N , s( M ) ) → a__U21#( a__and( a__isNat( M ) , isNat( N ) ) , M , N )

a__U21#( tt , M , N ) → a__plus#( mark( N ) , mark( M ) )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U11 (x₁, x₂) ] = 2 x₁

[mark (x₁) ] = x₁

[a__U21# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[U21 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 3

[a__plus (x₁, x₂) ] = 3 x₁ + 2 x₂

[0] = 0

[tt] = 0

[a__and (x₁, x₂) ] = x₁

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 2 x₁

[and (x₁, x₂) ] = x₁

[a__U21 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 3

[plus (x₁, x₂) ] = 3 x₁ + 2 x₂

[a__isNat (x₁) ] = 1

[s (x₁) ] = x₁ + 2

[a__plus# (x₁, x₂) ] = 2 x₁

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

one remains with the following pair(s).

a__U21#( tt , M , N ) → a__plus#( mark( N ) , mark( M ) )

1.1.1.1.1: dependency graph processor

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

The 2nd component contains the pair(s)

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

a__and#( tt , X ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

mark#( isNat( X ) ) → a__isNat#( X )

a__isNat#( plus( V1 , V2 ) ) → a__and#( a__isNat( V1 ) , isNat( V2 ) )

a__isNat#( plus( V1 , V2 ) ) → a__isNat#( V1 )

a__isNat#( s( V1 ) ) → a__isNat#( V1 )

mark#( s( X ) ) → mark#( X )

1.1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U11 (x₁, x₂) ] = x₁

[mark (x₁) ] = x₁

[U21 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 3

[a__and# (x₁, x₂) ] = x₁

[mark# (x₁) ] = x₁

[0] = 0

[a__plus (x₁, x₂) ] = 2 x₁ + 2 x₂

[tt] = 0

[a__isNat# (x₁) ] = 3 x₁

[a__and (x₁, x₂) ] = 2 x₁ + 2 x₂

[isNat (x₁) ] = 3 x₁

[U11 (x₁, x₂) ] = x₁

[a__U21 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 3

[and (x₁, x₂) ] = 2 x₁ + 2 x₂

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂

[s (x₁) ] = x₁ + 3

[a__isNat (x₁) ] = 3 x₁

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

one remains with the following pair(s).

1.1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U11 (x₁, x₂) ] = x₁

[mark (x₁) ] = x₁

[U21 (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[a__and# (x₁, x₂) ] = 2 x₁ + 3 x₂

[mark# (x₁) ] = 3 x₁ + 2

[0] = 2

[a__plus (x₁, x₂) ] = 3 x₁ + 3 x₂

[tt] = 1

[a__isNat# (x₁) ] = x₁ + 1

[a__and (x₁, x₂) ] = x₁ + x₂

[isNat (x₁) ] = x₁

[U11 (x₁, x₂) ] = x₁

[a__U21 (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[and (x₁, x₂) ] = x₁ + x₂

[plus (x₁, x₂) ] = 3 x₁ + 3 x₂

[s (x₁) ] = x₁

[a__isNat (x₁) ] = x₁

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

one remains with the following pair(s).

a__and#( tt , X ) → mark#( X )

mark#( and( X1 , X2 ) ) → mark#( X1 )

a__isNat#( plus( V1 , V2 ) ) → a__isNat#( V1 )

1.1.1.2.1.1: dependency graph processor

The dependency pairs are split into 2 component(s).

The 1st component contains the pair(s)

mark#( and( X1 , X2 ) ) → mark#( X1 )

1.1.1.2.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U11 (x₁, x₂) ] = x₁

[mark (x₁) ] = x₁

[U21 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 3

[mark# (x₁) ] = x₁

[a__plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 3

[0] = 0

[tt] = 0

[a__and (x₁, x₂) ] = x₁ + x₂ + 1

[isNat (x₁) ] = 3 x₁

[U11 (x₁, x₂) ] = x₁

[and (x₁, x₂) ] = x₁ + x₂ + 1

[a__U21 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 3

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 3

[a__isNat (x₁) ] = 3 x₁

[s (x₁) ] = x₁

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

one remains with the following pair(s).

none

1.1.1.2.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

a__isNat#( plus( V1 , V2 ) ) → a__isNat#( V1 )

1.1.1.2.1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U11 (x₁, x₂) ] = 2 x₁

[mark (x₁) ] = 2 x₁

[U21 (x₁, x₂, x₃) ] = x₁

[a__plus (x₁, x₂) ] = 2 x₁ + 1

[0] = 0

[a__isNat# (x₁) ] = x₁

[tt] = 0

[a__and (x₁, x₂) ] = 3 x₁

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = x₁

[and (x₁, x₂) ] = 2 x₁

[a__U21 (x₁, x₂, x₃) ] = 2 x₁

[plus (x₁, x₂) ] = 2 x₁ + 1

[a__isNat (x₁) ] = 0

[s (x₁) ] = 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

1.1.1.2.1.1.2.1: P is empty

All dependency pairs have been removed.

a__U11#( tt , N )	→	mark#( N )
a__U21#( tt , M , N )	→	a__plus#( mark( N ) , mark( M ) )
a__U21#( tt , M , N )	→	mark#( N )
a__U21#( tt , M , N )	→	mark#( M )
a__and#( tt , X )	→	mark#( X )
a__isNat#( plus( V1 , V2 ) )	→	a__and#( a__isNat( V1 ) , isNat( V2 ) )
a__isNat#( plus( V1 , V2 ) )	→	a__isNat#( V1 )
a__isNat#( s( V1 ) )	→	a__isNat#( V1 )
a__plus#( N , 0 )	→	a__U11#( a__isNat( N ) , N )
a__plus#( N , 0 )	→	a__isNat#( N )
a__plus#( N , s( M ) )	→	a__U21#( a__and( a__isNat( M ) , isNat( N ) ) , M , N )
a__plus#( N , s( M ) )	→	a__and#( a__isNat( M ) , isNat( N ) )
a__plus#( N , s( M ) )	→	a__isNat#( M )
mark#( U11( X1 , X2 ) )	→	a__U11#( mark( X1 ) , X2 )
mark#( U11( X1 , X2 ) )	→	mark#( X1 )
mark#( U21( X1 , X2 , X3 ) )	→	a__U21#( mark( X1 ) , X2 , X3 )
mark#( U21( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( plus( X1 , X2 ) )	→	a__plus#( mark( X1 ) , mark( X2 ) )
mark#( plus( X1 , X2 ) )	→	mark#( X1 )
mark#( plus( X1 , X2 ) )	→	mark#( X2 )
mark#( and( X1 , X2 ) )	→	a__and#( mark( X1 ) , X2 )
mark#( and( X1 , X2 ) )	→	mark#( X1 )
mark#( isNat( X ) )	→	a__isNat#( X )
mark#( s( X ) )	→	mark#( X )

[a__U11 (x₁, x₂) ]	=	2 x₁ + x₂ + 3
[mark (x₁) ]	=	x₁
[a__U21# (x₁, x₂, x₃) ]	=	x₁ + x₂ + 1
[U21 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + x₃ + 3
[a__and# (x₁, x₂) ]	=	2 x₁ + 1
[mark# (x₁) ]	=	x₁ + 1
[a__plus (x₁, x₂) ]	=	x₁ + x₂ + 3
[0]	=	3
[tt]	=	0
[a__isNat# (x₁) ]	=	1
[a__and (x₁, x₂) ]	=	x₁ + 2 x₂
[isNat (x₁) ]	=	0
[U11 (x₁, x₂) ]	=	2 x₁ + x₂ + 3
[and (x₁, x₂) ]	=	x₁ + 2 x₂
[a__U21 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + x₃ + 3
[plus (x₁, x₂) ]	=	x₁ + x₂ + 3
[a__U11# (x₁, x₂) ]	=	x₁ + 3
[s (x₁) ]	=	x₁
[a__isNat (x₁) ]	=	0
[a__plus# (x₁, x₂) ]	=	x₁ + x₂ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[a__U11 (x₁, x₂) ]	=	2 x₁
[mark (x₁) ]	=	x₁
[a__U21# (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂
[U21 (x₁, x₂, x₃) ]	=	2 x₁ + 3 x₂ + 3
[a__plus (x₁, x₂) ]	=	3 x₁ + 2 x₂
[0]	=	0
[tt]	=	0
[a__and (x₁, x₂) ]	=	x₁
[isNat (x₁) ]	=	1
[U11 (x₁, x₂) ]	=	2 x₁
[and (x₁, x₂) ]	=	x₁
[a__U21 (x₁, x₂, x₃) ]	=	2 x₁ + 3 x₂ + 3
[plus (x₁, x₂) ]	=	3 x₁ + 2 x₂
[a__isNat (x₁) ]	=	1
[s (x₁) ]	=	x₁ + 2
[a__plus# (x₁, x₂) ]	=	2 x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n