Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

U11#( tt , V1 , V2 ) → U12#( isNatKind( activate( V1 ) ) , activate( V1 ) , activate( V2 ) )

U11#( tt , V1 , V2 ) → isNatKind#( activate( V1 ) )

U11#( tt , V1 , V2 ) → activate#( V1 )

U11#( tt , V1 , V2 ) → activate#( V1 )

U11#( tt , V1 , V2 ) → activate#( V2 )

U12#( tt , V1 , V2 ) → U13#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )

U12#( tt , V1 , V2 ) → isNatKind#( activate( V2 ) )

U12#( tt , V1 , V2 ) → activate#( V2 )

U12#( tt , V1 , V2 ) → activate#( V1 )

U12#( tt , V1 , V2 ) → activate#( V2 )

U13#( tt , V1 , V2 ) → U14#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )

U13#( tt , V1 , V2 ) → isNatKind#( activate( V2 ) )

U13#( tt , V1 , V2 ) → activate#( V2 )

U13#( tt , V1 , V2 ) → activate#( V1 )

U13#( tt , V1 , V2 ) → activate#( V2 )

U14#( tt , V1 , V2 ) → U15#( isNat( activate( V1 ) ) , activate( V2 ) )

U14#( tt , V1 , V2 ) → isNat#( activate( V1 ) )

U14#( tt , V1 , V2 ) → activate#( V1 )

U14#( tt , V1 , V2 ) → activate#( V2 )

U15#( tt , V2 ) → U16#( isNat( activate( V2 ) ) )

U15#( tt , V2 ) → isNat#( activate( V2 ) )

U15#( tt , V2 ) → activate#( V2 )

U21#( tt , V1 ) → U22#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U21#( tt , V1 ) → isNatKind#( activate( V1 ) )

U21#( tt , V1 ) → activate#( V1 )

U21#( tt , V1 ) → activate#( V1 )

U22#( tt , V1 ) → U23#( isNat( activate( V1 ) ) )

U22#( tt , V1 ) → isNat#( activate( V1 ) )

U22#( tt , V1 ) → activate#( V1 )

U31#( tt , V2 ) → U32#( isNatKind( activate( V2 ) ) )

U31#( tt , V2 ) → isNatKind#( activate( V2 ) )

U31#( tt , V2 ) → activate#( V2 )

U51#( tt , N ) → U52#( isNatKind( activate( N ) ) , activate( N ) )

U51#( tt , N ) → isNatKind#( activate( N ) )

U51#( tt , N ) → activate#( N )

U51#( tt , N ) → activate#( N )

U52#( tt , N ) → activate#( N )

U61#( tt , M , N ) → U62#( isNatKind( activate( M ) ) , activate( M ) , activate( N ) )

U61#( tt , M , N ) → isNatKind#( activate( M ) )

U61#( tt , M , N ) → activate#( M )

U61#( tt , M , N ) → activate#( M )

U61#( tt , M , N ) → activate#( N )

U62#( tt , M , N ) → U63#( isNat( activate( N ) ) , activate( M ) , activate( N ) )

U62#( tt , M , N ) → isNat#( activate( N ) )

U62#( tt , M , N ) → activate#( N )

U62#( tt , M , N ) → activate#( M )

U62#( tt , M , N ) → activate#( N )

U63#( tt , M , N ) → U64#( isNatKind( activate( N ) ) , activate( M ) , activate( N ) )

U63#( tt , M , N ) → isNatKind#( activate( N ) )

U63#( tt , M , N ) → activate#( N )

U63#( tt , M , N ) → activate#( M )

U63#( tt , M , N ) → activate#( N )

U64#( tt , M , N ) → s#( plus( activate( N ) , activate( M ) ) )

U64#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

U64#( tt , M , N ) → activate#( N )

U64#( tt , M , N ) → activate#( M )

isNat#( n__plus( V1 , V2 ) ) → U11#( isNatKind( activate( V1 ) ) , activate( V1 ) , activate( V2 ) )

isNat#( n__plus( V1 , V2 ) ) → isNatKind#( activate( V1 ) )

isNat#( n__plus( V1 , V2 ) ) → activate#( V1 )

isNat#( n__plus( V1 , V2 ) ) → activate#( V1 )

isNat#( n__plus( V1 , V2 ) ) → activate#( V2 )

isNat#( n__s( V1 ) ) → U21#( isNatKind( activate( V1 ) ) , activate( V1 ) )

isNat#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

isNat#( n__s( V1 ) ) → activate#( V1 )

isNat#( n__s( V1 ) ) → activate#( V1 )

isNatKind#( n__plus( V1 , V2 ) ) → U31#( isNatKind( activate( V1 ) ) , activate( V2 ) )

isNatKind#( n__plus( V1 , V2 ) ) → isNatKind#( activate( V1 ) )

isNatKind#( n__plus( V1 , V2 ) ) → activate#( V1 )

isNatKind#( n__plus( V1 , V2 ) ) → activate#( V2 )

isNatKind#( n__s( V1 ) ) → U41#( isNatKind( activate( V1 ) ) )

isNatKind#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

isNatKind#( n__s( V1 ) ) → activate#( V1 )

plus#( N , 0 ) → U51#( isNat( N ) , N )

plus#( N , 0 ) → isNat#( N )

plus#( N , s( M ) ) → U61#( isNat( M ) , M , 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__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)

U12#( tt , V1 , V2 ) → U13#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )

U13#( tt , V1 , V2 ) → U14#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )

U14#( tt , V1 , V2 ) → U15#( isNat( activate( V1 ) ) , activate( V2 ) )

U15#( tt , V2 ) → isNat#( activate( V2 ) )

isNat#( n__plus( V1 , V2 ) ) → U11#( isNatKind( activate( V1 ) ) , activate( V1 ) , activate( V2 ) )

U11#( tt , V1 , V2 ) → U12#( isNatKind( activate( V1 ) ) , activate( V1 ) , activate( V2 ) )

U12#( tt , V1 , V2 ) → isNatKind#( activate( V2 ) )

isNatKind#( n__plus( V1 , V2 ) ) → U31#( isNatKind( activate( V1 ) ) , activate( V2 ) )

U31#( tt , V2 ) → isNatKind#( activate( V2 ) )

isNatKind#( n__plus( V1 , V2 ) ) → isNatKind#( activate( V1 ) )

isNatKind#( n__plus( V1 , V2 ) ) → activate#( V1 )

activate#( n__plus( X1 , X2 ) ) → plus#( activate( X1 ) , activate( X2 ) )

plus#( N , 0 ) → U51#( isNat( N ) , N )

U51#( tt , N ) → U52#( isNatKind( activate( N ) ) , activate( N ) )

U52#( tt , N ) → activate#( N )

activate#( n__plus( X1 , X2 ) ) → activate#( X1 )

activate#( n__plus( X1 , X2 ) ) → activate#( X2 )

activate#( n__s( X ) ) → activate#( X )

U51#( tt , N ) → isNatKind#( activate( N ) )

isNatKind#( n__plus( V1 , V2 ) ) → activate#( V2 )

isNatKind#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

isNatKind#( n__s( V1 ) ) → activate#( V1 )

U51#( tt , N ) → activate#( N )

plus#( N , 0 ) → isNat#( N )

isNat#( n__plus( V1 , V2 ) ) → isNatKind#( activate( V1 ) )

isNat#( n__plus( V1 , V2 ) ) → activate#( V1 )

isNat#( n__plus( V1 , V2 ) ) → activate#( V2 )

isNat#( n__s( V1 ) ) → U21#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U21#( tt , V1 ) → U22#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U22#( tt , V1 ) → isNat#( activate( V1 ) )

isNat#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

isNat#( n__s( V1 ) ) → activate#( V1 )

U22#( tt , V1 ) → activate#( V1 )

U21#( tt , V1 ) → isNatKind#( activate( V1 ) )

U21#( tt , V1 ) → activate#( V1 )

plus#( N , s( M ) ) → U61#( isNat( M ) , M , N )

U61#( tt , M , N ) → U62#( isNatKind( activate( M ) ) , activate( M ) , activate( N ) )

U62#( tt , M , N ) → U63#( isNat( activate( N ) ) , activate( M ) , activate( N ) )

U63#( tt , M , N ) → U64#( isNatKind( activate( N ) ) , activate( M ) , activate( N ) )

U64#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

plus#( N , s( M ) ) → isNat#( M )

U64#( tt , M , N ) → activate#( N )

U64#( tt , M , N ) → activate#( M )

U63#( tt , M , N ) → isNatKind#( activate( N ) )

U63#( tt , M , N ) → activate#( N )

U63#( tt , M , N ) → activate#( M )

U62#( tt , M , N ) → isNat#( activate( N ) )

U62#( tt , M , N ) → activate#( N )

U62#( tt , M , N ) → activate#( M )

U61#( tt , M , N ) → isNatKind#( activate( M ) )

U61#( tt , M , N ) → activate#( M )

U61#( tt , M , N ) → activate#( N )

U31#( tt , V2 ) → activate#( V2 )

U12#( tt , V1 , V2 ) → activate#( V2 )

U12#( tt , V1 , V2 ) → activate#( V1 )

U11#( tt , V1 , V2 ) → isNatKind#( activate( V1 ) )

U11#( tt , V1 , V2 ) → activate#( V1 )

U11#( tt , V1 , V2 ) → activate#( V2 )

U15#( tt , V2 ) → activate#( V2 )

U14#( tt , V1 , V2 ) → isNat#( activate( V1 ) )

U14#( tt , V1 , V2 ) → activate#( V1 )

U14#( tt , V1 , V2 ) → activate#( V2 )

U13#( tt , V1 , V2 ) → isNatKind#( activate( V2 ) )

U13#( tt , V1 , V2 ) → activate#( V2 )

U13#( tt , V1 , V2 ) → activate#( V1 )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[n__plus (x₁, x₂) ] = x₁ + 2 x₂ + 3

[U11 (x₁, x₂, x₃) ] = 2 x₁ + 2

[isNatKind (x₁) ] = 0

[U14# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 1

[n__0] = 1

[U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 3

[0] = 1

[U62# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[U61# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[U22# (x₁, x₂) ] = 3 x₁

[isNat# (x₁) ] = 3 x₁

[activate (x₁) ] = x₁

[U15 (x₁, x₂) ] = x₁

[U21 (x₁, x₂) ] = 0

[isNatKind# (x₁) ] = 3 x₁

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 0

[U64# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[U32 (x₁) ] = 0

[U13# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 1

[U12 (x₁, x₂, x₃) ] = 2 x₁ + 2

[s (x₁) ] = x₁

[U14 (x₁, x₂, x₃) ] = 2 x₁

[U12# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 1

[U52# (x₁, x₂) ] = 3 x₁

[U21# (x₁, x₂) ] = 3 x₁

[U31# (x₁, x₂) ] = 3 x₁

[U13 (x₁, x₂, x₃) ] = 2 x₁

[U11# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 2

[U52 (x₁, x₂) ] = x₁

[n__s (x₁) ] = x₁

[U51 (x₁, x₂) ] = x₁ + 2

[U63 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 3

[plus# (x₁, x₂) ] = 3 x₁ + 3 x₂

[tt] = 0

[isNat (x₁) ] = x₁

[U16 (x₁) ] = 0

[U41 (x₁) ] = 0

[U51# (x₁, x₂) ] = 3 x₁

[activate# (x₁) ] = 3 x₁

[U62 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 3

[plus (x₁, x₂) ] = x₁ + 2 x₂ + 3

[U23 (x₁) ] = 0

[U63# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[U15# (x₁, x₂) ] = 3 x₁

[U64 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 3

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

one remains with the following pair(s).

U12#( tt , V1 , V2 ) → U13#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )

U13#( tt , V1 , V2 ) → U14#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )

U15#( tt , V2 ) → isNat#( activate( V2 ) )

U31#( tt , V2 ) → isNatKind#( activate( V2 ) )

U51#( tt , N ) → U52#( isNatKind( activate( N ) ) , activate( N ) )

U52#( tt , N ) → activate#( N )

activate#( n__s( X ) ) → activate#( X )

U51#( tt , N ) → isNatKind#( activate( N ) )

isNatKind#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

isNatKind#( n__s( V1 ) ) → activate#( V1 )

U51#( tt , N ) → activate#( N )

isNat#( n__s( V1 ) ) → U21#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U21#( tt , V1 ) → U22#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U22#( tt , V1 ) → isNat#( activate( V1 ) )

isNat#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

isNat#( n__s( V1 ) ) → activate#( V1 )

U22#( tt , V1 ) → activate#( V1 )

U21#( tt , V1 ) → isNatKind#( activate( V1 ) )

U21#( tt , V1 ) → activate#( V1 )

plus#( N , s( M ) ) → U61#( isNat( M ) , M , N )

U61#( tt , M , N ) → U62#( isNatKind( activate( M ) ) , activate( M ) , activate( N ) )

U62#( tt , M , N ) → U63#( isNat( activate( N ) ) , activate( M ) , activate( N ) )

U63#( tt , M , N ) → U64#( isNatKind( activate( N ) ) , activate( M ) , activate( N ) )

U64#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

plus#( N , s( M ) ) → isNat#( M )

U64#( tt , M , N ) → activate#( N )

U64#( tt , M , N ) → activate#( M )

U63#( tt , M , N ) → isNatKind#( activate( N ) )

U63#( tt , M , N ) → activate#( N )

U63#( tt , M , N ) → activate#( M )

U62#( tt , M , N ) → isNat#( activate( N ) )

U62#( tt , M , N ) → activate#( N )

U62#( tt , M , N ) → activate#( M )

U61#( tt , M , N ) → isNatKind#( activate( M ) )

U61#( tt , M , N ) → activate#( M )

U61#( tt , M , N ) → activate#( N )

U31#( tt , V2 ) → activate#( V2 )

U15#( tt , V2 ) → activate#( V2 )

1.1.1.1: dependency graph processor

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

The 1st component contains the pair(s)

plus#( N , s( M ) ) → U61#( isNat( M ) , M , N )

U61#( tt , M , N ) → U62#( isNatKind( activate( M ) ) , activate( M ) , activate( N ) )

U62#( tt , M , N ) → U63#( isNat( activate( N ) ) , activate( M ) , activate( N ) )

U63#( tt , M , N ) → U64#( isNatKind( activate( N ) ) , activate( M ) , activate( N ) )

U64#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 0

[n__plus (x₁, x₂) ] = 3 x₁ + 2 x₂

[isNatKind (x₁) ] = 0

[n__0] = 0

[U61 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 2

[0] = 0

[U62# (x₁, x₂, x₃) ] = x₁ + 1

[U61# (x₁, x₂, x₃) ] = x₁ + 1

[activate (x₁) ] = x₁

[U15 (x₁, x₂) ] = 0

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 0

[U64# (x₁, x₂, x₃) ] = x₁

[U32 (x₁) ] = 0

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = x₁ + 1

[U14 (x₁, x₂, x₃) ] = 0

[U13 (x₁, x₂, x₃) ] = 0

[U52 (x₁, x₂) ] = x₁

[n__s (x₁) ] = x₁ + 1

[U63 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 1

[U51 (x₁, x₂) ] = x₁

[plus# (x₁, x₂) ] = x₁

[tt] = 0

[U16 (x₁) ] = 0

[isNat (x₁) ] = 0

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 2 x₁ + 3 x₂ + 2

[plus (x₁, x₂) ] = 3 x₁ + 2 x₂

[U23 (x₁) ] = 0

[U63# (x₁, x₂, x₃) ] = x₁ + 1

[U64 (x₁, x₂, x₃) ] = 2 x₁ + 3 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).

plus#( N , s( M ) ) → U61#( isNat( M ) , M , N )

U61#( tt , M , N ) → U62#( isNatKind( activate( M ) ) , activate( M ) , activate( N ) )

U62#( tt , M , N ) → U63#( isNat( activate( N ) ) , activate( M ) , activate( N ) )

U64#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

1.1.1.1.1.1: dependency graph processor

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

The 2nd component contains the pair(s)

isNat#( n__s( V1 ) ) → U21#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U21#( tt , V1 ) → U22#( isNatKind( activate( V1 ) ) , activate( V1 ) )

U22#( tt , V1 ) → isNat#( activate( V1 ) )

1.1.1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 1

[n__plus (x₁, x₂) ] = 2 x₁ + 2 x₂

[isNatKind (x₁) ] = 0

[n__0] = 0

[U61 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 2

[0] = 0

[isNat# (x₁) ] = 2 x₁ + 1

[U22# (x₁, x₂) ] = 2 x₁ + 2

[activate (x₁) ] = x₁

[U15 (x₁, x₂) ] = 0

[U21 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 1

[U32 (x₁) ] = 0

[s (x₁) ] = x₁ + 1

[U12 (x₁, x₂, x₃) ] = 0

[U14 (x₁, x₂, x₃) ] = 0

[U21# (x₁, x₂) ] = 2 x₁ + 3

[U13 (x₁, x₂, x₃) ] = 0

[U52 (x₁, x₂) ] = x₁

[n__s (x₁) ] = x₁ + 1

[U51 (x₁, x₂) ] = 2 x₁

[U63 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 1

[tt] = 0

[U16 (x₁) ] = 0

[isNat (x₁) ] = 2

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 1

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂

[U23 (x₁) ] = 1

[U64 (x₁, x₂, x₃) ] = 2 x₁ + 2 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).

isNat#( n__s( V1 ) ) → U21#( isNatKind( activate( V1 ) ) , activate( V1 ) )

1.1.1.1.2.1: dependency graph processor

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

The 3rd component contains the pair(s)

isNatKind#( n__s( V1 ) ) → isNatKind#( activate( V1 ) )

1.1.1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 0

[n__plus (x₁, x₂) ] = 2 x₁ + 2 x₂

[isNatKind (x₁) ] = 0

[n__0] = 0

[U61 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 2 x₃ + 3

[0] = 0

[activate (x₁) ] = x₁

[U15 (x₁, x₂) ] = 0

[isNatKind# (x₁) ] = 2 x₁

[U21 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 0

[U32 (x₁) ] = 0

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = x₁ + 2

[U14 (x₁, x₂, x₃) ] = 0

[U13 (x₁, x₂, x₃) ] = 0

[U52 (x₁, x₂) ] = x₁

[n__s (x₁) ] = x₁ + 2

[U51 (x₁, x₂) ] = x₁

[U63 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 2

[tt] = 0

[U16 (x₁) ] = 0

[isNat (x₁) ] = 1

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 3

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂

[U23 (x₁) ] = 0

[U64 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 2

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

one remains with the following pair(s).

none

1.1.1.1.3.1: P is empty

All dependency pairs have been removed.

The 4th component contains the pair(s)

activate#( n__s( X ) ) → activate#( X )

1.1.1.1.4: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 0

[n__plus (x₁, x₂) ] = x₁ + 2 x₂

[isNatKind (x₁) ] = 0

[n__0] = 2

[U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[0] = 2

[activate (x₁) ] = x₁

[U15 (x₁, x₂) ] = 0

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 0

[U32 (x₁) ] = 0

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = x₁ + 2

[U14 (x₁, x₂, x₃) ] = 0

[U13 (x₁, x₂, x₃) ] = 0

[U52 (x₁, x₂) ] = x₁

[n__s (x₁) ] = x₁ + 2

[U51 (x₁, x₂) ] = x₁

[U63 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[tt] = 0

[U16 (x₁) ] = 0

[isNat (x₁) ] = 0

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[activate# (x₁) ] = 2 x₁

[plus (x₁, x₂) ] = x₁ + 2 x₂

[U23 (x₁) ] = 0

[U64 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

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

one remains with the following pair(s).

none

1.1.1.1.4.1: P is empty

All dependency pairs have been removed.

U11#( tt , V1 , V2 )	→	U12#( isNatKind( activate( V1 ) ) , activate( V1 ) , activate( V2 ) )
U11#( tt , V1 , V2 )	→	isNatKind#( activate( V1 ) )
U11#( tt , V1 , V2 )	→	activate#( V1 )
U11#( tt , V1 , V2 )	→	activate#( V1 )
U11#( tt , V1 , V2 )	→	activate#( V2 )
U12#( tt , V1 , V2 )	→	U13#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )
U12#( tt , V1 , V2 )	→	isNatKind#( activate( V2 ) )
U12#( tt , V1 , V2 )	→	activate#( V2 )
U12#( tt , V1 , V2 )	→	activate#( V1 )
U12#( tt , V1 , V2 )	→	activate#( V2 )
U13#( tt , V1 , V2 )	→	U14#( isNatKind( activate( V2 ) ) , activate( V1 ) , activate( V2 ) )
U13#( tt , V1 , V2 )	→	isNatKind#( activate( V2 ) )
U13#( tt , V1 , V2 )	→	activate#( V2 )
U13#( tt , V1 , V2 )	→	activate#( V1 )
U13#( tt , V1 , V2 )	→	activate#( V2 )
U14#( tt , V1 , V2 )	→	U15#( isNat( activate( V1 ) ) , activate( V2 ) )
U14#( tt , V1 , V2 )	→	isNat#( activate( V1 ) )
U14#( tt , V1 , V2 )	→	activate#( V1 )
U14#( tt , V1 , V2 )	→	activate#( V2 )
U15#( tt , V2 )	→	U16#( isNat( activate( V2 ) ) )
U15#( tt , V2 )	→	isNat#( activate( V2 ) )
U15#( tt , V2 )	→	activate#( V2 )
U21#( tt , V1 )	→	U22#( isNatKind( activate( V1 ) ) , activate( V1 ) )
U21#( tt , V1 )	→	isNatKind#( activate( V1 ) )
U21#( tt , V1 )	→	activate#( V1 )
U21#( tt , V1 )	→	activate#( V1 )
U22#( tt , V1 )	→	U23#( isNat( activate( V1 ) ) )
U22#( tt , V1 )	→	isNat#( activate( V1 ) )
U22#( tt , V1 )	→	activate#( V1 )
U31#( tt , V2 )	→	U32#( isNatKind( activate( V2 ) ) )
U31#( tt , V2 )	→	isNatKind#( activate( V2 ) )
U31#( tt , V2 )	→	activate#( V2 )
U51#( tt , N )	→	U52#( isNatKind( activate( N ) ) , activate( N ) )
U51#( tt , N )	→	isNatKind#( activate( N ) )
U51#( tt , N )	→	activate#( N )
U51#( tt , N )	→	activate#( N )
U52#( tt , N )	→	activate#( N )
U61#( tt , M , N )	→	U62#( isNatKind( activate( M ) ) , activate( M ) , activate( N ) )
U61#( tt , M , N )	→	isNatKind#( activate( M ) )
U61#( tt , M , N )	→	activate#( M )
U61#( tt , M , N )	→	activate#( M )
U61#( tt , M , N )	→	activate#( N )
U62#( tt , M , N )	→	U63#( isNat( activate( N ) ) , activate( M ) , activate( N ) )
U62#( tt , M , N )	→	isNat#( activate( N ) )
U62#( tt , M , N )	→	activate#( N )
U62#( tt , M , N )	→	activate#( M )
U62#( tt , M , N )	→	activate#( N )
U63#( tt , M , N )	→	U64#( isNatKind( activate( N ) ) , activate( M ) , activate( N ) )
U63#( tt , M , N )	→	isNatKind#( activate( N ) )
U63#( tt , M , N )	→	activate#( N )
U63#( tt , M , N )	→	activate#( M )
U63#( tt , M , N )	→	activate#( N )
U64#( tt , M , N )	→	s#( plus( activate( N ) , activate( M ) ) )
U64#( tt , M , N )	→	plus#( activate( N ) , activate( M ) )
U64#( tt , M , N )	→	activate#( N )
U64#( tt , M , N )	→	activate#( M )
isNat#( n__plus( V1 , V2 ) )	→	U11#( isNatKind( activate( V1 ) ) , activate( V1 ) , activate( V2 ) )
isNat#( n__plus( V1 , V2 ) )	→	isNatKind#( activate( V1 ) )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V1 )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V1 )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V2 )
isNat#( n__s( V1 ) )	→	U21#( isNatKind( activate( V1 ) ) , activate( V1 ) )
isNat#( n__s( V1 ) )	→	isNatKind#( activate( V1 ) )
isNat#( n__s( V1 ) )	→	activate#( V1 )
isNat#( n__s( V1 ) )	→	activate#( V1 )
isNatKind#( n__plus( V1 , V2 ) )	→	U31#( isNatKind( activate( V1 ) ) , activate( V2 ) )
isNatKind#( n__plus( V1 , V2 ) )	→	isNatKind#( activate( V1 ) )
isNatKind#( n__plus( V1 , V2 ) )	→	activate#( V1 )
isNatKind#( n__plus( V1 , V2 ) )	→	activate#( V2 )
isNatKind#( n__s( V1 ) )	→	U41#( isNatKind( activate( V1 ) ) )
isNatKind#( n__s( V1 ) )	→	isNatKind#( activate( V1 ) )
isNatKind#( n__s( V1 ) )	→	activate#( V1 )
plus#( N , 0 )	→	U51#( isNat( N ) , N )
plus#( N , 0 )	→	isNat#( N )
plus#( N , s( M ) )	→	U61#( isNat( M ) , M , 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__s( X ) )	→	s#( activate( X ) )
activate#( n__s( X ) )	→	activate#( X )

[n__plus (x₁, x₂) ]	=	x₁ + 2 x₂ + 3
[U11 (x₁, x₂, x₃) ]	=	2 x₁ + 2
[isNatKind (x₁) ]	=	0
[U14# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂ + 1
[n__0]	=	1
[U61 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 3
[0]	=	1
[U62# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂
[U61# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂
[U22# (x₁, x₂) ]	=	3 x₁
[isNat# (x₁) ]	=	3 x₁
[activate (x₁) ]	=	x₁
[U15 (x₁, x₂) ]	=	x₁
[U21 (x₁, x₂) ]	=	0
[isNatKind# (x₁) ]	=	3 x₁
[U31 (x₁, x₂) ]	=	0
[U22 (x₁, x₂) ]	=	0
[U64# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂
[U32 (x₁) ]	=	0
[U13# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂ + 1
[U12 (x₁, x₂, x₃) ]	=	2 x₁ + 2
[s (x₁) ]	=	x₁
[U14 (x₁, x₂, x₃) ]	=	2 x₁
[U12# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂ + 1
[U52# (x₁, x₂) ]	=	3 x₁
[U21# (x₁, x₂) ]	=	3 x₁
[U31# (x₁, x₂) ]	=	3 x₁
[U13 (x₁, x₂, x₃) ]	=	2 x₁
[U11# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂ + 2
[U52 (x₁, x₂) ]	=	x₁
[n__s (x₁) ]	=	x₁
[U51 (x₁, x₂) ]	=	x₁ + 2
[U63 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 3
[plus# (x₁, x₂) ]	=	3 x₁ + 3 x₂
[tt]	=	0
[isNat (x₁) ]	=	x₁
[U16 (x₁) ]	=	0
[U41 (x₁) ]	=	0
[U51# (x₁, x₂) ]	=	3 x₁
[activate# (x₁) ]	=	3 x₁
[U62 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 3
[plus (x₁, x₂) ]	=	x₁ + 2 x₂ + 3
[U23 (x₁) ]	=	0
[U63# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂
[U15# (x₁, x₂) ]	=	3 x₁
[U64 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 3
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U11 (x₁, x₂, x₃) ]	=	0
[n__plus (x₁, x₂) ]	=	3 x₁ + 2 x₂
[isNatKind (x₁) ]	=	0
[n__0]	=	0
[U61 (x₁, x₂, x₃) ]	=	2 x₁ + 3 x₂ + 2
[0]	=	0
[U62# (x₁, x₂, x₃) ]	=	x₁ + 1
[U61# (x₁, x₂, x₃) ]	=	x₁ + 1
[activate (x₁) ]	=	x₁
[U15 (x₁, x₂) ]	=	0
[U21 (x₁, x₂) ]	=	0
[U31 (x₁, x₂) ]	=	0
[U22 (x₁, x₂) ]	=	0
[U64# (x₁, x₂, x₃) ]	=	x₁
[U32 (x₁) ]	=	0
[U12 (x₁, x₂, x₃) ]	=	0
[s (x₁) ]	=	x₁ + 1
[U14 (x₁, x₂, x₃) ]	=	0
[U13 (x₁, x₂, x₃) ]	=	0
[U52 (x₁, x₂) ]	=	x₁
[n__s (x₁) ]	=	x₁ + 1
[U63 (x₁, x₂, x₃) ]	=	2 x₁ + 3 x₂ + 1
[U51 (x₁, x₂) ]	=	x₁
[plus# (x₁, x₂) ]	=	x₁
[tt]	=	0
[U16 (x₁) ]	=	0
[isNat (x₁) ]	=	0
[U41 (x₁) ]	=	0
[U62 (x₁, x₂, x₃) ]	=	2 x₁ + 3 x₂ + 2
[plus (x₁, x₂) ]	=	3 x₁ + 2 x₂
[U23 (x₁) ]	=	0
[U63# (x₁, x₂, x₃) ]	=	x₁ + 1
[U64 (x₁, x₂, x₃) ]	=	2 x₁ + 3 x₂ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U11 (x₁, x₂, x₃) ]	=	1
[n__plus (x₁, x₂) ]	=	2 x₁ + 2 x₂
[isNatKind (x₁) ]	=	0
[n__0]	=	0
[U61 (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 2
[0]	=	0
[isNat# (x₁) ]	=	2 x₁ + 1
[U22# (x₁, x₂) ]	=	2 x₁ + 2
[activate (x₁) ]	=	x₁
[U15 (x₁, x₂) ]	=	0
[U21 (x₁, x₂) ]	=	1
[U31 (x₁, x₂) ]	=	0
[U22 (x₁, x₂) ]	=	1
[U32 (x₁) ]	=	0
[s (x₁) ]	=	x₁ + 1
[U12 (x₁, x₂, x₃) ]	=	0
[U14 (x₁, x₂, x₃) ]	=	0
[U21# (x₁, x₂) ]	=	2 x₁ + 3
[U13 (x₁, x₂, x₃) ]	=	0
[U52 (x₁, x₂) ]	=	x₁
[n__s (x₁) ]	=	x₁ + 1
[U51 (x₁, x₂) ]	=	2 x₁
[U63 (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 1
[tt]	=	0
[U16 (x₁) ]	=	0
[isNat (x₁) ]	=	2
[U41 (x₁) ]	=	0
[U62 (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 1
[plus (x₁, x₂) ]	=	2 x₁ + 2 x₂
[U23 (x₁) ]	=	1
[U64 (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U11 (x₁, x₂, x₃) ]	=	0
[n__plus (x₁, x₂) ]	=	x₁ + 2 x₂
[isNatKind (x₁) ]	=	0
[n__0]	=	2
[U61 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2
[0]	=	2
[activate (x₁) ]	=	x₁
[U15 (x₁, x₂) ]	=	0
[U21 (x₁, x₂) ]	=	0
[U31 (x₁, x₂) ]	=	0
[U22 (x₁, x₂) ]	=	0
[U32 (x₁) ]	=	0
[U12 (x₁, x₂, x₃) ]	=	0
[s (x₁) ]	=	x₁ + 2
[U14 (x₁, x₂, x₃) ]	=	0
[U13 (x₁, x₂, x₃) ]	=	0
[U52 (x₁, x₂) ]	=	x₁
[n__s (x₁) ]	=	x₁ + 2
[U51 (x₁, x₂) ]	=	x₁
[U63 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2
[tt]	=	0
[U16 (x₁) ]	=	0
[isNat (x₁) ]	=	0
[U41 (x₁) ]	=	0
[U62 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2
[activate# (x₁) ]	=	2 x₁
[plus (x₁, x₂) ]	=	x₁ + 2 x₂
[U23 (x₁) ]	=	0
[U64 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n