Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

active#( U11( tt , V2 ) ) → U12#( isNat( V2 ) )

active#( U11( tt , V2 ) ) → isNat#( V2 )

active#( U12( tt ) ) → mark#( tt )

active#( U21( tt ) ) → mark#( tt )

active#( U31( tt , N ) ) → mark#( N )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U41( tt , M , N ) ) → U42#( isNat( N ) , M , N )

active#( U41( tt , M , N ) ) → isNat#( N )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

active#( U42( tt , M , N ) ) → s#( plus( N , M ) )

active#( U42( tt , M , N ) ) → plus#( N , M )

active#( isNat( 0 ) ) → mark#( tt )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

active#( isNat( plus( V1 , V2 ) ) ) → U11#( isNat( V1 ) , V2 )

active#( isNat( plus( V1 , V2 ) ) ) → isNat#( V1 )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

active#( isNat( s( V1 ) ) ) → U21#( isNat( V1 ) )

active#( isNat( s( V1 ) ) ) → isNat#( V1 )

active#( plus( N , 0 ) ) → mark#( U31( isNat( N ) , N ) )

active#( plus( N , 0 ) ) → U31#( isNat( N ) , N )

active#( plus( N , 0 ) ) → isNat#( N )

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

active#( plus( N , s( M ) ) ) → U41#( isNat( M ) , M , N )

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

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

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

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( tt ) → active#( tt )

mark#( U12( X ) ) → active#( U12( mark( X ) ) )

mark#( U12( X ) ) → U12#( mark( X ) )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

mark#( isNat( X ) ) → isNat#( X )

mark#( U21( X ) ) → active#( U21( mark( X ) ) )

mark#( U21( X ) ) → U21#( mark( X ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → active#( U31( mark( X1 ) , X2 ) )

mark#( U31( X1 , X2 ) ) → U31#( mark( X1 ) , X2 )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

mark#( U41( X1 , X2 , X3 ) ) → U41#( mark( X1 ) , X2 , X3 )

mark#( U41( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

mark#( U42( X1 , X2 , X3 ) ) → U42#( mark( X1 ) , X2 , X3 )

mark#( U42( X1 , X2 , X3 ) ) → mark#( X1 )

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

mark#( s( X ) ) → s#( mark( X ) )

mark#( s( X ) ) → mark#( X )

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 )

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

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

U11#( active( X1 ) , X2 ) → U11#( X1 , X2 )

U11#( X1 , active( X2 ) ) → U11#( X1 , X2 )

U12#( mark( X ) ) → U12#( X )

U12#( active( X ) ) → U12#( X )

isNat#( mark( X ) ) → isNat#( X )

isNat#( active( X ) ) → isNat#( X )

U21#( mark( X ) ) → U21#( X )

U21#( active( X ) ) → U21#( X )

U31#( mark( X1 ) , X2 ) → U31#( X1 , X2 )

U31#( X1 , mark( X2 ) ) → U31#( X1 , X2 )

U31#( active( X1 ) , X2 ) → U31#( X1 , X2 )

U31#( X1 , active( X2 ) ) → U31#( X1 , X2 )

U41#( mark( X1 ) , X2 , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , mark( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , X2 , mark( X3 ) ) → U41#( X1 , X2 , X3 )

U41#( active( X1 ) , X2 , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , active( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , X2 , active( X3 ) ) → U41#( X1 , X2 , X3 )

U42#( mark( X1 ) , X2 , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , mark( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , X2 , mark( X3 ) ) → U42#( X1 , X2 , X3 )

U42#( active( X1 ) , X2 , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , active( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , X2 , active( X3 ) ) → U42#( X1 , X2 , X3 )

s#( mark( X ) ) → s#( X )

s#( active( X ) ) → s#( X )

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 )

1.1: dependency graph processor

The dependency pairs are split into 10 component(s).

The 1st component contains the pair(s)

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → active#( U12( mark( X ) ) )

active#( U31( tt , N ) ) → mark#( N )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

mark#( U21( X ) ) → active#( U21( mark( X ) ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → active#( U31( mark( X1 ) , X2 ) )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U41( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

active#( plus( N , 0 ) ) → mark#( U31( isNat( N ) , N ) )

mark#( U42( X1 , X2 , X3 ) ) → mark#( X1 )

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

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

mark#( s( X ) ) → mark#( X )

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

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

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

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → active#( U12( mark( X ) ) )

active#( U31( tt , N ) ) → mark#( N )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → active#( U31( mark( X1 ) , X2 ) )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U41( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

active#( plus( N , 0 ) ) → mark#( U31( isNat( N ) , N ) )

mark#( U42( X1 , X2 , X3 ) ) → mark#( X1 )

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

mark#( s( X ) ) → mark#( X )

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

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

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

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 0

[mark# (x₁) ] = 2

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

active#( U31( tt , N ) ) → mark#( N )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → active#( U31( mark( X1 ) , X2 ) )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U41( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

active#( plus( N , 0 ) ) → mark#( U31( isNat( N ) , N ) )

mark#( U42( X1 , X2 , X3 ) ) → mark#( X1 )

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

mark#( s( X ) ) → mark#( X )

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

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

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

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 2

[U41 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁ + 2 x₂

[U21 (x₁) ] = 2 x₁

[plus (x₁, x₂) ] = 2 x₁ + x₂

[s (x₁) ] = x₁

[U42 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃

[U12 (x₁) ] = 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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

active#( U31( tt , N ) ) → mark#( N )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → active#( U31( mark( X1 ) , X2 ) )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U41( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

mark#( U42( X1 , X2 , X3 ) ) → mark#( X1 )

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

mark#( s( X ) ) → mark#( X )

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

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

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

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = 3 x₁ + 1

[active (x₁) ] = x₁

[mark# (x₁) ] = 3 x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + x₂ + x₃ + 3

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 2 x₁

[U31 (x₁, x₂) ] = x₁ + x₂ + 1

[U21 (x₁) ] = 3 x₁

[plus (x₁, x₂) ] = x₁ + x₂ + 3

[s (x₁) ] = x₁

[U42 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + x₃ + 3

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → active#( U31( mark( X1 ) , X2 ) )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

mark#( s( X ) ) → mark#( X )

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

1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[active (x₁) ] = 0

[mark# (x₁) ] = 2

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2

[tt] = 0

[isNat (x₁) ] = 2

[U11 (x₁, x₂) ] = 2

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 2

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 2

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

active#( plus( N , s( M ) ) ) → mark#( U41( isNat( M ) , M , N ) )

mark#( s( X ) ) → mark#( X )

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

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 2 x₃

[tt] = 2

[isNat (x₁) ] = 2

[U11 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = x₁

[plus (x₁, x₂) ] = 2 x₁ + 3 x₂ + 2

[s (x₁) ] = x₁ + 2

[U42 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 2 x₃

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

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

1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 2

[active# (x₁) ] = x₁

[active (x₁) ] = 2

[mark# (x₁) ] = 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U41( tt , M , N ) ) → mark#( U42( isNat( N ) , M , N ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

mark#( U41( X1 , X2 , X3 ) ) → active#( U41( mark( X1 ) , X2 , X3 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = 0

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 3

[U41 (x₁, x₂, x₃) ] = 2

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + x₂ + 2

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (x₁) ] = 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).

1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = 2 x₁

[active (x₁) ] = 0

[mark# (x₁) ] = 0

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

active#( U42( tt , M , N ) ) → mark#( s( plus( N , M ) ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = 2 x₁

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[tt] = 3

[isNat (x₁) ] = 3

[U11 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁ + 1

[U21 (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + 3 x₂ + 2

[s (x₁) ] = 2

[U42 (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

mark#( U42( X1 , X2 , X3 ) ) → active#( U42( mark( X1 ) , X2 , X3 ) )

1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( U11( X1 , X2 ) ) → mark#( X1 )

mark#( U12( X ) ) → mark#( X )

mark#( isNat( X ) ) → active#( isNat( X ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( plus( V1 , V2 ) ) ) → mark#( U11( isNat( V1 ) , V2 ) )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[tt] = 0

[isNat (x₁) ] = 2 x₁

[U11 (x₁, x₂) ] = x₁ + 2 x₂ + 2

[U31 (x₁, x₂) ] = x₁

[U21 (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + x₂ + 2

[s (x₁) ] = x₁

[U42 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[U12 (x₁) ] = x₁ + 2

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

one remains with the following pair(s).

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

active#( U11( tt , V2 ) ) → mark#( U12( isNat( V2 ) ) )

mark#( isNat( X ) ) → active#( isNat( X ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = 1

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + 2

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + 2

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 2

[U12 (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).

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

mark#( isNat( X ) ) → active#( isNat( X ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 0

[active# (x₁) ] = x₁

[active (x₁) ] = 0

[mark# (x₁) ] = 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

mark#( isNat( X ) ) → active#( isNat( X ) )

mark#( U21( X ) ) → mark#( X )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active# (x₁) ] = x₁

[active (x₁) ] = x₁

[mark# (x₁) ] = x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[tt] = 0

[isNat (x₁) ] = x₁

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = x₁

[U21 (x₁) ] = x₁

[plus (x₁, x₂) ] = x₁ + x₂

[s (x₁) ] = x₁ + 1

[U42 (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[U12 (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).

mark#( U21( X ) ) → mark#( X )

active#( isNat( s( V1 ) ) ) → mark#( U21( isNat( V1 ) ) )

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: dependency graph processor

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

The 1st component contains the pair(s)

mark#( U21( X ) ) → mark#( X )

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁

[active (x₁) ] = x₁

[mark# (x₁) ] = 2 x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[tt] = 0

[isNat (x₁) ] = 2 x₁

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = x₁

[U21 (x₁) ] = x₁ + 1

[plus (x₁, x₂) ] = x₁ + 2 x₂

[s (x₁) ] = x₁ + 2

[U42 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[U12 (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

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

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

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

U11#( active( X1 ) , X2 ) → U11#( X1 , X2 )

U11#( X1 , active( X2 ) ) → U11#( X1 , X2 )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + 2

[tt] = 0

[isNat (x₁) ] = 2 x₁ + 2

[U11 (x₁, x₂) ] = 2

[U31 (x₁, x₂) ] = x₁ + 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U11# (x₁, x₂) ] = 3 x₁ + 3 x₂

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

U11#( active( X1 ) , X2 ) → U11#( X1 , X2 )

U11#( X1 , active( X2 ) ) → U11#( X1 , X2 )

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U11# (x₁, x₂) ] = 2 x₁ + 2 x₂

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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

1.1.2.1.1: P is empty

All dependency pairs have been removed.

The 3rd component contains the pair(s)

U12#( active( X ) ) → U12#( X )

U12#( mark( X ) ) → U12#( X )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2

[tt] = 0

[isNat (x₁) ] = x₁ + 2

[U11 (x₁, x₂) ] = 3

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U12# (x₁) ] = x₁

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 2

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

one remains with the following pair(s).

U12#( active( X ) ) → U12#( X )

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 1

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U12# (x₁) ] = 3 x₁

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

none

1.1.3.1.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

isNat#( active( X ) ) → isNat#( X )

isNat#( mark( X ) ) → isNat#( X )

1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2

[isNat# (x₁) ] = x₁

[tt] = 0

[isNat (x₁) ] = x₁ + 2

[U11 (x₁, x₂) ] = 3

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 2

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

one remains with the following pair(s).

isNat#( active( X ) ) → isNat#( X )

1.1.4.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[isNat# (x₁) ] = 3 x₁

[tt] = 1

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

[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.

The 5th component contains the pair(s)

U21#( active( X ) ) → U21#( X )

U21#( mark( X ) ) → U21#( X )

1.1.5: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U21# (x₁) ] = x₁

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2

[tt] = 0

[isNat (x₁) ] = x₁ + 2

[U11 (x₁, x₂) ] = 3

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 2

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

one remains with the following pair(s).

U21#( active( X ) ) → U21#( X )

1.1.5.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U21# (x₁) ] = 3 x₁

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 1

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

none

1.1.5.1.1: P is empty

All dependency pairs have been removed.

The 6th component contains the pair(s)

U31#( X1 , mark( X2 ) ) → U31#( X1 , X2 )

U31#( mark( X1 ) , X2 ) → U31#( X1 , X2 )

U31#( active( X1 ) , X2 ) → U31#( X1 , X2 )

U31#( X1 , active( X2 ) ) → U31#( X1 , X2 )

1.1.6: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + 2

[tt] = 0

[isNat (x₁) ] = 2 x₁ + 2

[U11 (x₁, x₂) ] = 2

[U31 (x₁, x₂) ] = x₁ + 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U31# (x₁, x₂) ] = 3 x₁ + 3 x₂

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

U31#( active( X1 ) , X2 ) → U31#( X1 , X2 )

U31#( X1 , active( X2 ) ) → U31#( X1 , X2 )

1.1.6.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U31# (x₁, x₂) ] = 2 x₁ + 2 x₂

[U12 (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

1.1.6.1.1: P is empty

All dependency pairs have been removed.

The 7th component contains the pair(s)

U41#( X1 , mark( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

U41#( mark( X1 ) , X2 , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , X2 , mark( X3 ) ) → U41#( X1 , X2 , X3 )

U41#( active( X1 ) , X2 , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , active( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , X2 , active( X3 ) ) → U41#( X1 , X2 , X3 )

1.1.7: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 3 x₃

[U41 (x₁, x₂, x₃) ] = x₁ + 2

[tt] = 0

[isNat (x₁) ] = x₁ + 2

[U11 (x₁, x₂) ] = 2

[U31 (x₁, x₂) ] = x₁ + 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = x₁ + 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

U41#( active( X1 ) , X2 , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , active( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , X2 , active( X3 ) ) → U41#( X1 , X2 , X3 )

1.1.7.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[U41# (x₁, x₂, x₃) ] = 2 x₁

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

U41#( X1 , active( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

U41#( X1 , X2 , active( X3 ) ) → U41#( X1 , X2 , X3 )

1.1.7.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = x₁ + 3

[0] = 0

[U41# (x₁, x₂, x₃) ] = 3 x₁

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

U41#( X1 , active( X2 ) , X3 ) → U41#( X1 , X2 , X3 )

1.1.7.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[U41# (x₁, x₂, x₃) ] = 3 x₁

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 0

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

none

1.1.7.1.1.1.1: P is empty

All dependency pairs have been removed.

The 8th component contains the pair(s)

U42#( X1 , mark( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

U42#( mark( X1 ) , X2 , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , X2 , mark( X3 ) ) → U42#( X1 , X2 , X3 )

U42#( active( X1 ) , X2 , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , active( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , X2 , active( X3 ) ) → U42#( X1 , X2 , X3 )

1.1.8: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + 2

[tt] = 0

[U42# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 3 x₃

[isNat (x₁) ] = x₁ + 2

[U11 (x₁, x₂) ] = 2

[U31 (x₁, x₂) ] = x₁ + 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = x₁ + 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

U42#( active( X1 ) , X2 , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , active( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , X2 , active( X3 ) ) → U42#( X1 , X2 , X3 )

1.1.8.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[U42# (x₁, x₂, x₃) ] = 2 x₁

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

U42#( X1 , active( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

U42#( X1 , X2 , active( X3 ) ) → U42#( X1 , X2 , X3 )

1.1.8.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = x₁ + 3

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[tt] = 0

[U42# (x₁, x₂, x₃) ] = 3 x₁

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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).

U42#( X1 , active( X2 ) , X3 ) → U42#( X1 , X2 , X3 )

1.1.8.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[tt] = 0

[U42# (x₁, x₂, x₃) ] = 3 x₁

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

none

1.1.8.1.1.1.1: P is empty

All dependency pairs have been removed.

The 9th component contains the pair(s)

s#( active( X ) ) → s#( X )

s#( mark( X ) ) → s#( X )

1.1.9: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = 2

[s# (x₁) ] = x₁

[tt] = 0

[isNat (x₁) ] = x₁ + 2

[U11 (x₁, x₂) ] = 3

[U31 (x₁, x₂) ] = x₁ + 2

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 2

[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 )

1.1.9.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 1

[0] = 0

[U41 (x₁, x₂, x₃) ] = 1

[s# (x₁) ] = 3 x₁

[tt] = 1

[isNat (x₁) ] = 1

[U11 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = 1

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

none

1.1.9.1.1: P is empty

All dependency pairs have been removed.

The 10th 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 )

1.1.10: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = x₁ + 1

[active (x₁) ] = x₁

[0] = 0

[U41 (x₁, x₂, x₃) ] = x₁ + 2

[plus# (x₁, x₂) ] = 3 x₁ + 3 x₂

[tt] = 0

[isNat (x₁) ] = 2 x₁ + 2

[U11 (x₁, x₂) ] = 2

[U31 (x₁, x₂) ] = x₁ + 1

[U21 (x₁) ] = 1

[plus (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 1

[U12 (x₁) ] = 1

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

one remains with the following pair(s).

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

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

1.1.10.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[mark (x₁) ] = 3

[active (x₁) ] = 2 x₁ + 3

[0] = 0

[U41 (x₁, x₂, x₃) ] = 0

[plus# (x₁, x₂) ] = 2 x₁ + 2 x₂

[tt] = 0

[isNat (x₁) ] = 0

[U11 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U21 (x₁) ] = 0

[plus (x₁, x₂) ] = 0

[s (x₁) ] = 0

[U42 (x₁, x₂, x₃) ] = 0

[U12 (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

1.1.10.1.1: P is empty

All dependency pairs have been removed.

active#( U11( tt , V2 ) )	→	mark#( U12( isNat( V2 ) ) )
active#( U11( tt , V2 ) )	→	U12#( isNat( V2 ) )
active#( U11( tt , V2 ) )	→	isNat#( V2 )
active#( U12( tt ) )	→	mark#( tt )
active#( U21( tt ) )	→	mark#( tt )
active#( U31( tt , N ) )	→	mark#( N )
active#( U41( tt , M , N ) )	→	mark#( U42( isNat( N ) , M , N ) )
active#( U41( tt , M , N ) )	→	U42#( isNat( N ) , M , N )
active#( U41( tt , M , N ) )	→	isNat#( N )
active#( U42( tt , M , N ) )	→	mark#( s( plus( N , M ) ) )
active#( U42( tt , M , N ) )	→	s#( plus( N , M ) )
active#( U42( tt , M , N ) )	→	plus#( N , M )
active#( isNat( 0 ) )	→	mark#( tt )
active#( isNat( plus( V1 , V2 ) ) )	→	mark#( U11( isNat( V1 ) , V2 ) )
active#( isNat( plus( V1 , V2 ) ) )	→	U11#( isNat( V1 ) , V2 )
active#( isNat( plus( V1 , V2 ) ) )	→	isNat#( V1 )
active#( isNat( s( V1 ) ) )	→	mark#( U21( isNat( V1 ) ) )
active#( isNat( s( V1 ) ) )	→	U21#( isNat( V1 ) )
active#( isNat( s( V1 ) ) )	→	isNat#( V1 )
active#( plus( N , 0 ) )	→	mark#( U31( isNat( N ) , N ) )
active#( plus( N , 0 ) )	→	U31#( isNat( N ) , N )
active#( plus( N , 0 ) )	→	isNat#( N )
active#( plus( N , s( M ) ) )	→	mark#( U41( isNat( M ) , M , N ) )
active#( plus( N , s( M ) ) )	→	U41#( isNat( M ) , M , N )
active#( plus( N , s( M ) ) )	→	isNat#( M )
mark#( U11( X1 , X2 ) )	→	active#( U11( mark( X1 ) , X2 ) )
mark#( U11( X1 , X2 ) )	→	U11#( mark( X1 ) , X2 )
mark#( U11( X1 , X2 ) )	→	mark#( X1 )
mark#( tt )	→	active#( tt )
mark#( U12( X ) )	→	active#( U12( mark( X ) ) )
mark#( U12( X ) )	→	U12#( mark( X ) )
mark#( U12( X ) )	→	mark#( X )
mark#( isNat( X ) )	→	active#( isNat( X ) )
mark#( isNat( X ) )	→	isNat#( X )
mark#( U21( X ) )	→	active#( U21( mark( X ) ) )
mark#( U21( X ) )	→	U21#( mark( X ) )
mark#( U21( X ) )	→	mark#( X )
mark#( U31( X1 , X2 ) )	→	active#( U31( mark( X1 ) , X2 ) )
mark#( U31( X1 , X2 ) )	→	U31#( mark( X1 ) , X2 )
mark#( U31( X1 , X2 ) )	→	mark#( X1 )
mark#( U41( X1 , X2 , X3 ) )	→	active#( U41( mark( X1 ) , X2 , X3 ) )
mark#( U41( X1 , X2 , X3 ) )	→	U41#( mark( X1 ) , X2 , X3 )
mark#( U41( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U42( X1 , X2 , X3 ) )	→	active#( U42( mark( X1 ) , X2 , X3 ) )
mark#( U42( X1 , X2 , X3 ) )	→	U42#( mark( X1 ) , X2 , X3 )
mark#( U42( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( s( X ) )	→	active#( s( mark( X ) ) )
mark#( s( X ) )	→	s#( mark( X ) )
mark#( s( X ) )	→	mark#( X )
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 )
U11#( mark( X1 ) , X2 )	→	U11#( X1 , X2 )
U11#( X1 , mark( X2 ) )	→	U11#( X1 , X2 )
U11#( active( X1 ) , X2 )	→	U11#( X1 , X2 )
U11#( X1 , active( X2 ) )	→	U11#( X1 , X2 )
U12#( mark( X ) )	→	U12#( X )
U12#( active( X ) )	→	U12#( X )
isNat#( mark( X ) )	→	isNat#( X )
isNat#( active( X ) )	→	isNat#( X )
U21#( mark( X ) )	→	U21#( X )
U21#( active( X ) )	→	U21#( X )
U31#( mark( X1 ) , X2 )	→	U31#( X1 , X2 )
U31#( X1 , mark( X2 ) )	→	U31#( X1 , X2 )
U31#( active( X1 ) , X2 )	→	U31#( X1 , X2 )
U31#( X1 , active( X2 ) )	→	U31#( X1 , X2 )
U41#( mark( X1 ) , X2 , X3 )	→	U41#( X1 , X2 , X3 )
U41#( X1 , mark( X2 ) , X3 )	→	U41#( X1 , X2 , X3 )
U41#( X1 , X2 , mark( X3 ) )	→	U41#( X1 , X2 , X3 )
U41#( active( X1 ) , X2 , X3 )	→	U41#( X1 , X2 , X3 )
U41#( X1 , active( X2 ) , X3 )	→	U41#( X1 , X2 , X3 )
U41#( X1 , X2 , active( X3 ) )	→	U41#( X1 , X2 , X3 )
U42#( mark( X1 ) , X2 , X3 )	→	U42#( X1 , X2 , X3 )
U42#( X1 , mark( X2 ) , X3 )	→	U42#( X1 , X2 , X3 )
U42#( X1 , X2 , mark( X3 ) )	→	U42#( X1 , X2 , X3 )
U42#( active( X1 ) , X2 , X3 )	→	U42#( X1 , X2 , X3 )
U42#( X1 , active( X2 ) , X3 )	→	U42#( X1 , X2 , X3 )
U42#( X1 , X2 , active( X3 ) )	→	U42#( X1 , X2 , X3 )
s#( mark( X ) )	→	s#( X )
s#( active( X ) )	→	s#( X )
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 )

[mark (x₁) ]	=	0
[active# (x₁) ]	=	x₁
[active (x₁) ]	=	0
[mark# (x₁) ]	=	1
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	1
[tt]	=	0
[isNat (x₁) ]	=	1
[U11 (x₁, x₂) ]	=	1
[U31 (x₁, x₂) ]	=	1
[U21 (x₁) ]	=	0
[plus (x₁, x₂) ]	=	1
[s (x₁) ]	=	0
[U42 (x₁, x₂, x₃) ]	=	1
[U12 (x₁) ]	=	1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	0
[active# (x₁) ]	=	2 x₁
[active (x₁) ]	=	0
[mark# (x₁) ]	=	2
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	1
[tt]	=	0
[isNat (x₁) ]	=	1
[U11 (x₁, x₂) ]	=	1
[U31 (x₁, x₂) ]	=	1
[U21 (x₁) ]	=	0
[plus (x₁, x₂) ]	=	1
[s (x₁) ]	=	0
[U42 (x₁, x₂, x₃) ]	=	1
[U12 (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	x₁ + 1
[active (x₁) ]	=	x₁
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	x₁ + 2
[tt]	=	0
[isNat (x₁) ]	=	2 x₁ + 2
[U11 (x₁, x₂) ]	=	2
[U31 (x₁, x₂) ]	=	x₁ + 1
[U21 (x₁) ]	=	1
[plus (x₁, x₂) ]	=	x₁ + 3
[s (x₁) ]	=	0
[U11# (x₁, x₂) ]	=	3 x₁ + 3 x₂
[U42 (x₁, x₂, x₃) ]	=	1
[U12 (x₁) ]	=	1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	3
[active (x₁) ]	=	2 x₁ + 3
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	0
[tt]	=	0
[isNat (x₁) ]	=	0
[U11 (x₁, x₂) ]	=	0
[U31 (x₁, x₂) ]	=	0
[U21 (x₁) ]	=	0
[plus (x₁, x₂) ]	=	0
[s (x₁) ]	=	0
[U11# (x₁, x₂) ]	=	2 x₁ + 2 x₂
[U42 (x₁, x₂, x₃) ]	=	0
[U12 (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[mark (x₁) ]	=	3
[active (x₁) ]	=	2 x₁ + 1
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	1
[tt]	=	1
[isNat (x₁) ]	=	1
[U11 (x₁, x₂) ]	=	1
[U31 (x₁, x₂) ]	=	1
[U21 (x₁) ]	=	1
[plus (x₁, x₂) ]	=	1
[s (x₁) ]	=	0
[U12# (x₁) ]	=	3 x₁
[U42 (x₁, x₂, x₃) ]	=	1
[U12 (x₁) ]	=	1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U21# (x₁) ]	=	x₁
[mark (x₁) ]	=	x₁ + 1
[active (x₁) ]	=	x₁
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	2
[tt]	=	0
[isNat (x₁) ]	=	x₁ + 2
[U11 (x₁, x₂) ]	=	3
[U31 (x₁, x₂) ]	=	x₁ + 2
[U21 (x₁) ]	=	1
[plus (x₁, x₂) ]	=	x₁ + 3
[s (x₁) ]	=	0
[U42 (x₁, x₂, x₃) ]	=	1
[U12 (x₁) ]	=	2
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n