Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

a__U11#( tt , V1 , V2 ) → a__U12#( a__isNatKind( V1 ) , V1 , V2 )

a__U11#( tt , V1 , V2 ) → a__isNatKind#( V1 )

a__U12#( tt , V1 , V2 ) → a__U13#( a__isNatKind( V2 ) , V1 , V2 )

a__U12#( tt , V1 , V2 ) → a__isNatKind#( V2 )

a__U13#( tt , V1 , V2 ) → a__U14#( a__isNatKind( V2 ) , V1 , V2 )

a__U13#( tt , V1 , V2 ) → a__isNatKind#( V2 )

a__U14#( tt , V1 , V2 ) → a__U15#( a__isNat( V1 ) , V2 )

a__U14#( tt , V1 , V2 ) → a__isNat#( V1 )

a__U15#( tt , V2 ) → a__U16#( a__isNat( V2 ) )

a__U15#( tt , V2 ) → a__isNat#( V2 )

a__U21#( tt , V1 ) → a__U22#( a__isNatKind( V1 ) , V1 )

a__U21#( tt , V1 ) → a__isNatKind#( V1 )

a__U22#( tt , V1 ) → a__U23#( a__isNat( V1 ) )

a__U22#( tt , V1 ) → a__isNat#( V1 )

a__U31#( tt , V2 ) → a__U32#( a__isNatKind( V2 ) )

a__U31#( tt , V2 ) → a__isNatKind#( V2 )

a__U51#( tt , N ) → a__U52#( a__isNatKind( N ) , N )

a__U51#( tt , N ) → a__isNatKind#( N )

a__U52#( tt , N ) → mark#( N )

a__U61#( tt , M , N ) → a__U62#( a__isNatKind( M ) , M , N )

a__U61#( tt , M , N ) → a__isNatKind#( M )

a__U62#( tt , M , N ) → a__U63#( a__isNat( N ) , M , N )

a__U62#( tt , M , N ) → a__isNat#( N )

a__U63#( tt , M , N ) → a__U64#( a__isNatKind( N ) , M , N )

a__U63#( tt , M , N ) → a__isNatKind#( N )

a__U64#( tt , M , N ) → a__plus#( mark( N ) , mark( M ) )

a__U64#( tt , M , N ) → mark#( N )

a__U64#( tt , M , N ) → mark#( M )

a__isNat#( plus( V1 , V2 ) ) → a__U11#( a__isNatKind( V1 ) , V1 , V2 )

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

a__isNat#( s( V1 ) ) → a__U21#( a__isNatKind( V1 ) , V1 )

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

a__isNatKind#( plus( V1 , V2 ) ) → a__U31#( a__isNatKind( V1 ) , V2 )

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

a__isNatKind#( s( V1 ) ) → a__U41#( a__isNatKind( V1 ) )

a__isNatKind#( s( V1 ) ) → a__isNatKind#( V1 )

a__plus#( N , 0 ) → a__U51#( a__isNat( N ) , N )

a__plus#( N , 0 ) → a__isNat#( N )

a__plus#( N , s( M ) ) → a__U61#( a__isNat( M ) , M , N )

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

mark#( U11( X1 , X2 , X3 ) ) → a__U11#( mark( X1 ) , X2 , X3 )

mark#( U11( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U12( X1 , X2 , X3 ) ) → a__U12#( mark( X1 ) , X2 , X3 )

mark#( U12( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( isNatKind( X ) ) → a__isNatKind#( X )

mark#( U13( X1 , X2 , X3 ) ) → a__U13#( mark( X1 ) , X2 , X3 )

mark#( U13( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U14( X1 , X2 , X3 ) ) → a__U14#( mark( X1 ) , X2 , X3 )

mark#( U14( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U15( X1 , X2 ) ) → a__U15#( mark( X1 ) , X2 )

mark#( U15( X1 , X2 ) ) → mark#( X1 )

mark#( isNat( X ) ) → a__isNat#( X )

mark#( U16( X ) ) → a__U16#( mark( X ) )

mark#( U16( X ) ) → mark#( X )

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

mark#( U21( X1 , X2 ) ) → mark#( X1 )

mark#( U22( X1 , X2 ) ) → a__U22#( mark( X1 ) , X2 )

mark#( U22( X1 , X2 ) ) → mark#( X1 )

mark#( U23( X ) ) → a__U23#( mark( X ) )

mark#( U23( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → a__U31#( mark( X1 ) , X2 )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U32( X ) ) → a__U32#( mark( X ) )

mark#( U32( X ) ) → mark#( X )

mark#( U41( X ) ) → a__U41#( mark( X ) )

mark#( U41( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → a__U51#( mark( X1 ) , X2 )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X1 , X2 ) ) → a__U52#( mark( X1 ) , X2 )

mark#( U52( X1 , X2 ) ) → mark#( X1 )

mark#( U61( X1 , X2 , X3 ) ) → a__U61#( mark( X1 ) , X2 , X3 )

mark#( U61( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U62( X1 , X2 , X3 ) ) → a__U62#( mark( X1 ) , X2 , X3 )

mark#( U62( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U63( X1 , X2 , X3 ) ) → a__U63#( mark( X1 ) , X2 , X3 )

mark#( U63( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U64( X1 , X2 , X3 ) ) → a__U64#( mark( X1 ) , X2 , X3 )

mark#( U64( 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#( s( X ) ) → mark#( X )

1.1: dependency graph processor

The dependency pairs are split into 3 component(s).

The 1st component contains the pair(s)

mark#( U11( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U12( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U13( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U14( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U15( X1 , X2 ) ) → mark#( X1 )

mark#( U16( X ) ) → mark#( X )

mark#( U21( X1 , X2 ) ) → mark#( X1 )

mark#( U22( X1 , X2 ) ) → mark#( X1 )

mark#( U23( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U32( X ) ) → mark#( X )

mark#( U41( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → a__U51#( mark( X1 ) , X2 )

a__U51#( tt , N ) → a__U52#( a__isNatKind( N ) , N )

a__U52#( tt , N ) → mark#( N )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X1 , X2 ) ) → a__U52#( mark( X1 ) , X2 )

mark#( U52( X1 , X2 ) ) → mark#( X1 )

mark#( U61( X1 , X2 , X3 ) ) → a__U61#( mark( X1 ) , X2 , X3 )

a__U61#( tt , M , N ) → a__U62#( a__isNatKind( M ) , M , N )

a__U62#( tt , M , N ) → a__U63#( a__isNat( N ) , M , N )

a__U63#( tt , M , N ) → a__U64#( a__isNatKind( N ) , M , N )

a__U64#( tt , M , N ) → a__plus#( mark( N ) , mark( M ) )

a__plus#( N , 0 ) → a__U51#( a__isNat( N ) , N )

a__plus#( N , s( M ) ) → a__U61#( a__isNat( M ) , M , N )

a__U64#( tt , M , N ) → mark#( N )

mark#( U61( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U62( X1 , X2 , X3 ) ) → a__U62#( mark( X1 ) , X2 , X3 )

mark#( U62( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U63( X1 , X2 , X3 ) ) → a__U63#( mark( X1 ) , X2 , X3 )

mark#( U63( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U64( X1 , X2 , X3 ) ) → a__U64#( mark( X1 ) , X2 , X3 )

a__U64#( tt , M , N ) → mark#( M )

mark#( U64( 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#( s( X ) ) → mark#( X )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁ + x₂ + 1

[a__U16 (x₁) ] = 3 x₁

[a__plus (x₁, x₂) ] = 2 x₁ + x₂

[0] = 1

[U21 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁

[a__U23 (x₁) ] = 2 x₁

[a__U12 (x₁, x₂, x₃) ] = 2 x₁

[a__U14 (x₁, x₂, x₃) ] = x₁

[U12 (x₁, x₂, x₃) ] = 2 x₁

[a__U52# (x₁, x₂) ] = 2 x₁

[a__U51# (x₁, x₂) ] = 2 x₁ + 2

[a__U21 (x₁, x₂) ] = x₁

[mark# (x₁) ] = 2 x₁

[a__U13 (x₁, x₂, x₃) ] = x₁

[U63 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃ + 1

[a__U41 (x₁) ] = 3 x₁

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃ + 1

[a__U22 (x₁, x₂) ] = 2 x₁

[a__U32 (x₁) ] = x₁

[U41 (x₁) ] = 3 x₁

[a__isNatKind (x₁) ] = 0

[a__U64# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[plus (x₁, x₂) ] = 2 x₁ + x₂

[U23 (x₁) ] = 2 x₁

[a__isNat (x₁) ] = 0

[a__U62# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 2

[a__plus# (x₁, x₂) ] = 2 x₁ + 2 x₂

[a__U63 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃ + 1

[U64 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃ + 1

[U11 (x₁, x₂, x₃) ] = x₁

[a__U63# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 1

[a__U64 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃ + 1

[U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃ + 1

[U15 (x₁, x₂) ] = x₁

[a__U31 (x₁, x₂) ] = x₁

[U22 (x₁, x₂) ] = 2 x₁

[a__U15 (x₁, x₂) ] = x₁

[U32 (x₁) ] = x₁

[s (x₁) ] = x₁ + 1

[U14 (x₁, x₂, x₃) ] = x₁

[a__U61# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + 2

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = x₁

[a__U52 (x₁, x₂) ] = 2 x₁ + x₂

[U52 (x₁, x₂) ] = 2 x₁ + x₂

[U51 (x₁, x₂) ] = x₁ + x₂ + 1

[a__U62 (x₁, x₂, x₃) ] = x₁ + x₂ + 2 x₃ + 1

[U16 (x₁) ] = 3 x₁

[isNat (x₁) ] = 0

[a__U11 (x₁, x₂, x₃) ] = x₁

[U62 (x₁, x₂, x₃) ] = x₁ + x₂ + 2 x₃ + 1

[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: dependency graph processor

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

The 1st component contains the pair(s)

mark#( U12( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U11( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U13( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U14( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U15( X1 , X2 ) ) → mark#( X1 )

mark#( U16( X ) ) → mark#( X )

mark#( U21( X1 , X2 ) ) → mark#( X1 )

mark#( U22( X1 , X2 ) ) → mark#( X1 )

mark#( U23( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U32( X ) ) → mark#( X )

mark#( U41( X ) ) → mark#( X )

mark#( U52( X1 , X2 ) ) → a__U52#( mark( X1 ) , X2 )

a__U52#( tt , N ) → mark#( N )

mark#( U52( X1 , X2 ) ) → mark#( X1 )

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

[U11 (x₁, x₂, x₃) ] = x₁

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = 3 x₁

[a__U64 (x₁, x₂, x₃) ] = 0

[U61 (x₁, x₂, x₃) ] = 1

[a__U16 (x₁) ] = x₁

[a__plus (x₁, x₂) ] = 3 x₁ + 2 x₂ + 2

[0] = 2

[U15 (x₁, x₂) ] = 2 x₁

[U21 (x₁, x₂) ] = 2 x₁

[a__U31 (x₁, x₂) ] = 2 x₁

[U31 (x₁, x₂) ] = 2 x₁

[U22 (x₁, x₂) ] = x₁

[a__U23 (x₁) ] = 2 x₁

[a__U12 (x₁, x₂, x₃) ] = x₁

[a__U15 (x₁, x₂) ] = 2 x₁

[U32 (x₁) ] = 2 x₁

[a__U14 (x₁, x₂, x₃) ] = 2 x₁

[s (x₁) ] = 0

[U12 (x₁, x₂, x₃) ] = x₁

[U14 (x₁, x₂, x₃) ] = 2 x₁

[a__U52# (x₁, x₂) ] = 2 x₁

[mark (x₁) ] = 2 x₁

[U13 (x₁, x₂, x₃) ] = 2 x₁

[a__U21 (x₁, x₂) ] = 2 x₁

[a__U52 (x₁, x₂) ] = 2 x₁ + 2 x₂

[U52 (x₁, x₂) ] = 2 x₁ + x₂

[mark# (x₁) ] = 2 x₁

[a__U13 (x₁, x₂, x₃) ] = 2 x₁

[U51 (x₁, x₂) ] = 3 x₁

[U63 (x₁, x₂, x₃) ] = 1

[tt] = 0

[a__U41 (x₁) ] = x₁

[a__U62 (x₁, x₂, x₃) ] = 2

[a__U61 (x₁, x₂, x₃) ] = 2

[U16 (x₁) ] = x₁

[isNat (x₁) ] = 0

[a__U11 (x₁, x₂, x₃) ] = x₁

[a__U22 (x₁, x₂) ] = x₁

[a__isNatKind (x₁) ] = 0

[U41 (x₁) ] = x₁

[a__U32 (x₁) ] = 2 x₁

[U62 (x₁, x₂, x₃) ] = 2

[plus (x₁, x₂) ] = 3 x₁ + 2 x₂ + 2

[U23 (x₁) ] = 2 x₁

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = 2

[U64 (x₁, x₂, x₃) ] = 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

[U11 (x₁, x₂, x₃) ] = x₁

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁ + 2

[a__U64 (x₁, x₂, x₃) ] = 0

[U61 (x₁, x₂, x₃) ] = 0

[a__U16 (x₁) ] = 2 x₁

[a__plus (x₁, x₂) ] = x₁ + 2

[0] = 0

[U15 (x₁, x₂) ] = 2 x₁

[U21 (x₁, x₂) ] = 2 x₁

[a__U31 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁

[U22 (x₁, x₂) ] = x₁

[a__U23 (x₁) ] = x₁

[a__U12 (x₁, x₂, x₃) ] = 2 x₁

[a__U15 (x₁, x₂) ] = 2 x₁

[U32 (x₁) ] = x₁

[a__U14 (x₁, x₂, x₃) ] = x₁

[s (x₁) ] = 0

[U12 (x₁, x₂, x₃) ] = 2 x₁

[U14 (x₁, x₂, x₃) ] = x₁

[a__U52# (x₁, x₂) ] = x₁

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = x₁

[a__U21 (x₁, x₂) ] = 2 x₁

[a__U52 (x₁, x₂) ] = 2 x₁ + x₂ + 2

[U52 (x₁, x₂) ] = 2 x₁ + x₂ + 2

[mark# (x₁) ] = x₁

[a__U13 (x₁, x₂, x₃) ] = x₁

[U51 (x₁, x₂) ] = x₁ + 2

[U63 (x₁, x₂, x₃) ] = 0

[tt] = 0

[a__U41 (x₁) ] = x₁

[a__U62 (x₁, x₂, x₃) ] = 0

[a__U61 (x₁, x₂, x₃) ] = 0

[U16 (x₁) ] = 2 x₁

[isNat (x₁) ] = 0

[a__U11 (x₁, x₂, x₃) ] = x₁

[a__U22 (x₁, x₂) ] = x₁

[a__isNatKind (x₁) ] = 0

[U41 (x₁) ] = x₁

[a__U32 (x₁) ] = x₁

[U62 (x₁, x₂, x₃) ] = 0

[plus (x₁, x₂) ] = x₁ + 2

[U23 (x₁) ] = x₁

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = 0

[U64 (x₁, x₂, x₃) ] = 0

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

one remains with the following pair(s).

1.1.1.1.1.1.1: dependency graph processor

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

The 1st component contains the pair(s)

1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 3 x₃ + 1

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁

[a__U64 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[U61 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[a__U16 (x₁) ] = 2 x₁

[a__plus (x₁, x₂) ] = x₁ + 3 x₂ + 1

[0] = 0

[U15 (x₁, x₂) ] = x₁ + 2 x₂

[U21 (x₁, x₂) ] = x₁ + x₂ + 2

[a__U31 (x₁, x₂) ] = x₁

[U31 (x₁, x₂) ] = x₁

[U22 (x₁, x₂) ] = x₁ + x₂ + 2

[a__U23 (x₁) ] = x₁ + 2

[a__U12 (x₁, x₂, x₃) ] = x₁ + x₂ + 3 x₃

[a__U15 (x₁, x₂) ] = x₁ + 2 x₂

[U32 (x₁) ] = x₁

[a__U14 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃

[s (x₁) ] = x₁ + 2

[U12 (x₁, x₂, x₃) ] = x₁ + x₂ + 3 x₃

[U14 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃

[a__U52 (x₁, x₂) ] = x₁

[a__U21 (x₁, x₂) ] = x₁ + x₂ + 2

[U52 (x₁, x₂) ] = x₁

[mark# (x₁) ] = 2 x₁

[a__U13 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2 x₃

[U51 (x₁, x₂) ] = x₁

[U63 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[tt] = 0

[a__U41 (x₁) ] = 2 x₁

[a__U62 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[a__U61 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[U16 (x₁) ] = 2 x₁

[isNat (x₁) ] = x₁

[a__U22 (x₁, x₂) ] = x₁ + x₂ + 2

[a__U11 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 3 x₃ + 1

[a__isNatKind (x₁) ] = 0

[U41 (x₁) ] = 2 x₁

[a__U32 (x₁) ] = x₁

[U62 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[plus (x₁, x₂) ] = x₁ + 3 x₂ + 1

[U23 (x₁) ] = x₁ + 2

[a__isNat (x₁) ] = x₁

[a__U63 (x₁, x₂, x₃) ] = 3 x₁ + x₂ + 3

[U64 (x₁, x₂, x₃) ] = 3 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).

mark#( U12( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U13( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U14( X1 , X2 , X3 ) ) → mark#( X1 )

mark#( U15( X1 , X2 ) ) → mark#( X1 )

mark#( U16( X ) ) → mark#( X )

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U32( X ) ) → mark#( X )

mark#( U41( X ) ) → mark#( X )

1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = 2 x₁ + 2

[a__U64 (x₁, x₂, x₃) ] = 0

[U61 (x₁, x₂, x₃) ] = 2

[a__U16 (x₁) ] = x₁ + 1

[a__plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 2

[0] = 0

[U15 (x₁, x₂) ] = 2 x₁ + x₂ + 1

[a__U31 (x₁, x₂) ] = x₁

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = x₁

[U22 (x₁, x₂) ] = 0

[a__U23 (x₁) ] = 0

[a__U12 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + x₃ + 1

[a__U15 (x₁, x₂) ] = 2 x₁ + x₂ + 1

[U32 (x₁) ] = x₁

[a__U14 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + x₃ + 1

[s (x₁) ] = 0

[U12 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + x₃ + 1

[U14 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + x₃ + 1

[mark (x₁) ] = 2 x₁

[U13 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + x₃ + 1

[a__U52 (x₁, x₂) ] = 2 x₁

[a__U21 (x₁, x₂) ] = 0

[U52 (x₁, x₂) ] = 2 x₁

[mark# (x₁) ] = 2 x₁

[a__U13 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂ + x₃ + 1

[U51 (x₁, x₂) ] = 2 x₁ + 1

[U63 (x₁, x₂, x₃) ] = 0

[a__U41 (x₁) ] = 2 x₁

[tt] = 0

[a__U62 (x₁, x₂, x₃) ] = 2

[a__U61 (x₁, x₂, x₃) ] = 2

[U16 (x₁) ] = x₁ + 1

[isNat (x₁) ] = x₁

[a__U22 (x₁, x₂) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 1

[a__U32 (x₁) ] = x₁

[U41 (x₁) ] = 2 x₁

[a__isNatKind (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 1

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[U23 (x₁) ] = 0

[a__isNat (x₁) ] = x₁

[a__U63 (x₁, x₂, x₃) ] = 0

[U64 (x₁, x₂, x₃) ] = 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#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U32( X ) ) → mark#( X )

mark#( U41( X ) ) → mark#( X )

1.1.1.1.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 0

[isNatKind (x₁) ] = 2 x₁

[a__U51 (x₁, x₂) ] = x₁

[a__U64 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = x₁ + 2 x₂

[0] = 0

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = x₁ + 3 x₂

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = x₁ + 3 x₂

[U22 (x₁, x₂) ] = 0

[a__U23 (x₁) ] = 0

[a__U15 (x₁, x₂) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 0

[a__U14 (x₁, x₂, x₃) ] = 0

[U32 (x₁) ] = x₁

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = x₁ + 2

[U14 (x₁, x₂, x₃) ] = 0

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = 0

[a__U21 (x₁, x₂) ] = 0

[a__U52 (x₁, x₂) ] = x₁

[U52 (x₁, x₂) ] = x₁

[mark# (x₁) ] = 3 x₁

[a__U13 (x₁, x₂, x₃) ] = 0

[U51 (x₁, x₂) ] = x₁

[U63 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[a__U62 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[a__U41 (x₁) ] = x₁ + 2

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[isNat (x₁) ] = 0

[U16 (x₁) ] = 0

[a__U22 (x₁, x₂) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 0

[a__U32 (x₁) ] = x₁

[U41 (x₁) ] = x₁ + 2

[a__isNatKind (x₁) ] = 2 x₁

[U62 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[plus (x₁, x₂) ] = x₁ + 2 x₂

[U23 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 2

[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).

mark#( U31( X1 , X2 ) ) → mark#( X1 )

mark#( U32( X ) ) → mark#( X )

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

[U11 (x₁, x₂, x₃) ] = 0

[isNatKind (x₁) ] = x₁

[a__U64 (x₁, x₂, x₃) ] = 0

[a__U51 (x₁, x₂) ] = x₁ + 3

[U61 (x₁, x₂, x₃) ] = 3 x₁ + 2

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = 3 x₁ + 3 x₂ + 2

[0] = 3

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = x₁ + 3 x₂ + 1

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = x₁ + 3 x₂ + 1

[U22 (x₁, x₂) ] = 0

[a__U23 (x₁) ] = x₁

[a__U15 (x₁, x₂) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 0

[a__U14 (x₁, x₂, x₃) ] = 0

[U32 (x₁) ] = 2 x₁

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = 0

[U14 (x₁, x₂, x₃) ] = 0

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = 0

[a__U21 (x₁, x₂) ] = 0

[a__U52 (x₁, x₂) ] = x₁

[U52 (x₁, x₂) ] = x₁

[mark# (x₁) ] = 2 x₁

[a__U13 (x₁, x₂, x₃) ] = 0

[U51 (x₁, x₂) ] = x₁ + 3

[U63 (x₁, x₂, x₃) ] = 0

[a__U62 (x₁, x₂, x₃) ] = 0

[a__U41 (x₁) ] = 0

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 3 x₁ + 2

[isNat (x₁) ] = 0

[U16 (x₁) ] = 0

[a__U22 (x₁, x₂) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 0

[a__U32 (x₁) ] = 2 x₁

[a__isNatKind (x₁) ] = x₁

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 0

[plus (x₁, x₂) ] = 3 x₁ + 3 x₂ + 2

[U23 (x₁) ] = x₁

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = 0

[U64 (x₁, x₂, x₃) ] = 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#( U32( X ) ) → mark#( X )

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

[U11 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[isNatKind (x₁) ] = x₁

[a__U64 (x₁, x₂, x₃) ] = 2 x₁ + 2

[a__U51 (x₁, x₂) ] = 2 x₁ + 2

[U61 (x₁, x₂, x₃) ] = x₁ + 2

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 2

[0] = 0

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[U21 (x₁, x₂) ] = 1

[U31 (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[U22 (x₁, x₂) ] = 1

[a__U23 (x₁) ] = 1

[a__U15 (x₁, x₂) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 2

[a__U14 (x₁, x₂, x₃) ] = 2

[U32 (x₁) ] = x₁ + 1

[U12 (x₁, x₂, x₃) ] = 2

[s (x₁) ] = 2

[U14 (x₁, x₂, x₃) ] = 1

[mark (x₁) ] = 2 x₁

[U13 (x₁, x₂, x₃) ] = 1

[a__U21 (x₁, x₂) ] = 2

[a__U52 (x₁, x₂) ] = 2 x₁

[U52 (x₁, x₂) ] = x₁

[mark# (x₁) ] = 2 x₁

[a__U13 (x₁, x₂, x₃) ] = 2

[U51 (x₁, x₂) ] = x₁ + 1

[U63 (x₁, x₂, x₃) ] = 2 x₁ + 2

[a__U62 (x₁, x₂, x₃) ] = 2 x₁ + 3

[a__U41 (x₁) ] = 0

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 2 x₁ + 3

[isNat (x₁) ] = 2 x₁ + 2

[U16 (x₁) ] = 0

[a__U11 (x₁, x₂, x₃) ] = x₁ + 3 x₂ + 2

[a__U22 (x₁, x₂) ] = 2

[a__U32 (x₁) ] = x₁ + 1

[a__isNatKind (x₁) ] = 2 x₁

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 2 x₁ + 2

[plus (x₁, x₂) ] = 2 x₁ + 2 x₂ + 1

[U23 (x₁) ] = 1

[a__isNat (x₁) ] = 3 x₁ + 2

[a__U63 (x₁, x₂, x₃) ] = 2 x₁ + 3

[U64 (x₁, x₂, x₃) ] = 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).

none

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)

a__U12#( tt , V1 , V2 ) → a__U13#( a__isNatKind( V2 ) , V1 , V2 )

a__U13#( tt , V1 , V2 ) → a__U14#( a__isNatKind( V2 ) , V1 , V2 )

a__U14#( tt , V1 , V2 ) → a__U15#( a__isNat( V1 ) , V2 )

a__U15#( tt , V2 ) → a__isNat#( V2 )

a__isNat#( plus( V1 , V2 ) ) → a__U11#( a__isNatKind( V1 ) , V1 , V2 )

a__U11#( tt , V1 , V2 ) → a__U12#( a__isNatKind( V1 ) , V1 , V2 )

a__isNat#( s( V1 ) ) → a__U21#( a__isNatKind( V1 ) , V1 )

a__U21#( tt , V1 ) → a__U22#( a__isNatKind( V1 ) , V1 )

a__U22#( tt , V1 ) → a__isNat#( V1 )

a__U14#( tt , V1 , V2 ) → a__isNat#( V1 )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = x₁ + x₂

[0] = 0

[a__U11# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[a__U22# (x₁, x₂) ] = 2 x₁ + 2

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[a__U23 (x₁) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 0

[a__U14 (x₁, x₂, x₃) ] = 0

[U12 (x₁, x₂, x₃) ] = 0

[a__U21 (x₁, x₂) ] = 0

[a__U13 (x₁, x₂, x₃) ] = 0

[U63 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[a__isNat# (x₁) ] = 2 x₁

[tt] = 0

[a__U41 (x₁) ] = 0

[a__U61 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[a__U22 (x₁, x₂) ] = 0

[a__U21# (x₁, x₂) ] = 2 x₁ + 3

[a__U14# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[a__isNatKind (x₁) ] = 0

[a__U32 (x₁) ] = 0

[U41 (x₁) ] = 0

[a__U15# (x₁, x₂) ] = 2 x₁

[plus (x₁, x₂) ] = x₁ + x₂

[a__U12# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[U23 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[U64 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[U11 (x₁, x₂, x₃) ] = 0

[a__U64 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[U61 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = 0

[a__U13# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[U22 (x₁, x₂) ] = 0

[a__U15 (x₁, x₂) ] = 0

[U32 (x₁) ] = 0

[s (x₁) ] = x₁ + 2

[U14 (x₁, x₂, x₃) ] = 0

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = 0

[a__U52 (x₁, x₂) ] = x₁ + x₂

[U52 (x₁, x₂) ] = x₁ + x₂

[U51 (x₁, x₂) ] = x₁

[a__U62 (x₁, x₂, x₃) ] = x₁ + x₂ + 2

[U16 (x₁) ] = 0

[isNat (x₁) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 0

[U62 (x₁, x₂, x₃) ] = 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).

1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = 2 x₁ + 3

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = 2 x₁ + x₂ + 2

[0] = 2

[a__U11# (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 1

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[a__U23 (x₁) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 3 x₁

[a__U14 (x₁, x₂, x₃) ] = 0

[U12 (x₁, x₂, x₃) ] = 3 x₁

[a__U21 (x₁, x₂) ] = 0

[a__U13 (x₁, x₂, x₃) ] = 0

[U63 (x₁, x₂, x₃) ] = 0

[a__isNat# (x₁) ] = x₁

[tt] = 0

[a__U41 (x₁) ] = 0

[a__U61 (x₁, x₂, x₃) ] = 2 x₁

[a__U22 (x₁, x₂) ] = 0

[a__U14# (x₁, x₂, x₃) ] = x₁ + x₂

[U41 (x₁) ] = 0

[a__isNatKind (x₁) ] = 0

[a__U32 (x₁) ] = 0

[a__U15# (x₁, x₂) ] = x₁

[plus (x₁, x₂) ] = 2 x₁ + x₂ + 2

[a__U12# (x₁, x₂, x₃) ] = 2 x₁ + x₂ + 1

[U23 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = 0

[U64 (x₁, x₂, x₃) ] = 0

[U11 (x₁, x₂, x₃) ] = 0

[a__U64 (x₁, x₂, x₃) ] = 0

[U61 (x₁, x₂, x₃) ] = x₁

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = 0

[a__U13# (x₁, x₂, x₃) ] = x₁ + x₂

[U22 (x₁, x₂) ] = 0

[a__U15 (x₁, x₂) ] = 0

[U32 (x₁) ] = 0

[s (x₁) ] = 0

[U14 (x₁, x₂, x₃) ] = 0

[mark (x₁) ] = 2 x₁

[U13 (x₁, x₂, x₃) ] = 0

[a__U52 (x₁, x₂) ] = 2 x₁ + 3

[U52 (x₁, x₂) ] = 2 x₁ + 2

[U51 (x₁, x₂) ] = 2 x₁ + 3

[a__U62 (x₁, x₂, x₃) ] = 0

[U16 (x₁) ] = 0

[isNat (x₁) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 0

[U62 (x₁, x₂, x₃) ] = 0

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

one remains with the following pair(s).

a__U13#( tt , V1 , V2 ) → a__U14#( a__isNatKind( V2 ) , V1 , V2 )

a__U14#( tt , V1 , V2 ) → a__U15#( a__isNat( V1 ) , V2 )

a__U15#( tt , V2 ) → a__isNat#( V2 )

a__U11#( tt , V1 , V2 ) → a__U12#( a__isNatKind( V1 ) , V1 , V2 )

a__U14#( tt , V1 , V2 ) → a__isNat#( V1 )

1.1.2.1.1: dependency graph processor

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

The 3rd component contains the pair(s)

a__U31#( tt , V2 ) → a__isNatKind#( V2 )

a__isNatKind#( plus( V1 , V2 ) ) → a__U31#( a__isNatKind( V1 ) , V2 )

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

a__isNatKind#( s( V1 ) ) → a__isNatKind#( V1 )

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 0

[isNatKind (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁

[a__U64 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[U61 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = 2 x₁ + x₂

[0] = 0

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = 0

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 0

[a__U31# (x₁, x₂) ] = 2 x₁

[a__U23 (x₁) ] = 0

[a__U15 (x₁, x₂) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 0

[U32 (x₁) ] = 0

[a__U14 (x₁, x₂, x₃) ] = 0

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = x₁ + 1

[U14 (x₁, x₂, x₃) ] = 0

[mark (x₁) ] = x₁

[U13 (x₁, x₂, x₃) ] = 0

[a__U52 (x₁, x₂) ] = x₁

[a__U21 (x₁, x₂) ] = 0

[U52 (x₁, x₂) ] = x₁

[a__U13 (x₁, x₂, x₃) ] = 0

[U51 (x₁, x₂) ] = x₁

[U63 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[a__U62 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[a__U41 (x₁) ] = 2 x₁

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[isNat (x₁) ] = 0

[U16 (x₁) ] = 0

[a__U22 (x₁, x₂) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 0

[a__U32 (x₁) ] = 0

[a__isNatKind (x₁) ] = 0

[U41 (x₁) ] = 2 x₁

[U62 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[plus (x₁, x₂) ] = 2 x₁ + x₂

[U23 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = x₁ + 2 x₂ + 1

[a__isNatKind# (x₁) ] = 2 x₁

[U64 (x₁, x₂, x₃) ] = 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).

a__U31#( tt , V2 ) → a__isNatKind#( V2 )

a__isNatKind#( plus( V1 , V2 ) ) → a__U31#( a__isNatKind( V1 ) , V2 )

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

1.1.3.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = 0

[isNatKind (x₁) ] = 0

[a__U64 (x₁, x₂, x₃) ] = 0

[a__U51 (x₁, x₂) ] = 2 x₁ + 2

[U61 (x₁, x₂, x₃) ] = x₁ + 1

[a__U16 (x₁) ] = 0

[a__plus (x₁, x₂) ] = 2 x₁ + x₂ + 1

[0] = 2

[U15 (x₁, x₂) ] = 0

[a__U31 (x₁, x₂) ] = 0

[U21 (x₁, x₂) ] = 0

[U31 (x₁, x₂) ] = 0

[U22 (x₁, x₂) ] = 0

[a__U31# (x₁, x₂) ] = x₁ + 1

[a__U23 (x₁) ] = 0

[a__U15 (x₁, x₂) ] = 0

[a__U12 (x₁, x₂, x₃) ] = 0

[U32 (x₁) ] = 0

[a__U14 (x₁, x₂, x₃) ] = 0

[U12 (x₁, x₂, x₃) ] = 0

[s (x₁) ] = 0

[U14 (x₁, x₂, x₃) ] = 0

[mark (x₁) ] = 2 x₁

[U13 (x₁, x₂, x₃) ] = 0

[a__U52 (x₁, x₂) ] = 2 x₁ + 2

[a__U21 (x₁, x₂) ] = 0

[U52 (x₁, x₂) ] = x₁ + 1

[a__U13 (x₁, x₂, x₃) ] = 0

[U51 (x₁, x₂) ] = 2 x₁ + 1

[U63 (x₁, x₂, x₃) ] = 0

[a__U62 (x₁, x₂, x₃) ] = 0

[a__U41 (x₁) ] = 0

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = x₁ + 1

[isNat (x₁) ] = 0

[U16 (x₁) ] = 0

[a__U22 (x₁, x₂) ] = 0

[a__U11 (x₁, x₂, x₃) ] = 0

[a__U32 (x₁) ] = 0

[a__isNatKind (x₁) ] = 0

[U41 (x₁) ] = 0

[U62 (x₁, x₂, x₃) ] = 0

[plus (x₁, x₂) ] = 2 x₁ + x₂ + 1

[U23 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U63 (x₁, x₂, x₃) ] = 0

[a__isNatKind# (x₁) ] = x₁

[U64 (x₁, x₂, x₃) ] = 0

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

one remains with the following pair(s).

a__isNatKind#( plus( V1 , V2 ) ) → a__U31#( a__isNatKind( V1 ) , V2 )

1.1.3.1.1: dependency graph processor

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

a__U11#( tt , V1 , V2 )	→	a__U12#( a__isNatKind( V1 ) , V1 , V2 )
a__U11#( tt , V1 , V2 )	→	a__isNatKind#( V1 )
a__U12#( tt , V1 , V2 )	→	a__U13#( a__isNatKind( V2 ) , V1 , V2 )
a__U12#( tt , V1 , V2 )	→	a__isNatKind#( V2 )
a__U13#( tt , V1 , V2 )	→	a__U14#( a__isNatKind( V2 ) , V1 , V2 )
a__U13#( tt , V1 , V2 )	→	a__isNatKind#( V2 )
a__U14#( tt , V1 , V2 )	→	a__U15#( a__isNat( V1 ) , V2 )
a__U14#( tt , V1 , V2 )	→	a__isNat#( V1 )
a__U15#( tt , V2 )	→	a__U16#( a__isNat( V2 ) )
a__U15#( tt , V2 )	→	a__isNat#( V2 )
a__U21#( tt , V1 )	→	a__U22#( a__isNatKind( V1 ) , V1 )
a__U21#( tt , V1 )	→	a__isNatKind#( V1 )
a__U22#( tt , V1 )	→	a__U23#( a__isNat( V1 ) )
a__U22#( tt , V1 )	→	a__isNat#( V1 )
a__U31#( tt , V2 )	→	a__U32#( a__isNatKind( V2 ) )
a__U31#( tt , V2 )	→	a__isNatKind#( V2 )
a__U51#( tt , N )	→	a__U52#( a__isNatKind( N ) , N )
a__U51#( tt , N )	→	a__isNatKind#( N )
a__U52#( tt , N )	→	mark#( N )
a__U61#( tt , M , N )	→	a__U62#( a__isNatKind( M ) , M , N )
a__U61#( tt , M , N )	→	a__isNatKind#( M )
a__U62#( tt , M , N )	→	a__U63#( a__isNat( N ) , M , N )
a__U62#( tt , M , N )	→	a__isNat#( N )
a__U63#( tt , M , N )	→	a__U64#( a__isNatKind( N ) , M , N )
a__U63#( tt , M , N )	→	a__isNatKind#( N )
a__U64#( tt , M , N )	→	a__plus#( mark( N ) , mark( M ) )
a__U64#( tt , M , N )	→	mark#( N )
a__U64#( tt , M , N )	→	mark#( M )
a__isNat#( plus( V1 , V2 ) )	→	a__U11#( a__isNatKind( V1 ) , V1 , V2 )
a__isNat#( plus( V1 , V2 ) )	→	a__isNatKind#( V1 )
a__isNat#( s( V1 ) )	→	a__U21#( a__isNatKind( V1 ) , V1 )
a__isNat#( s( V1 ) )	→	a__isNatKind#( V1 )
a__isNatKind#( plus( V1 , V2 ) )	→	a__U31#( a__isNatKind( V1 ) , V2 )
a__isNatKind#( plus( V1 , V2 ) )	→	a__isNatKind#( V1 )
a__isNatKind#( s( V1 ) )	→	a__U41#( a__isNatKind( V1 ) )
a__isNatKind#( s( V1 ) )	→	a__isNatKind#( V1 )
a__plus#( N , 0 )	→	a__U51#( a__isNat( N ) , N )
a__plus#( N , 0 )	→	a__isNat#( N )
a__plus#( N , s( M ) )	→	a__U61#( a__isNat( M ) , M , N )
a__plus#( N , s( M ) )	→	a__isNat#( M )
mark#( U11( X1 , X2 , X3 ) )	→	a__U11#( mark( X1 ) , X2 , X3 )
mark#( U11( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U12( X1 , X2 , X3 ) )	→	a__U12#( mark( X1 ) , X2 , X3 )
mark#( U12( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( isNatKind( X ) )	→	a__isNatKind#( X )
mark#( U13( X1 , X2 , X3 ) )	→	a__U13#( mark( X1 ) , X2 , X3 )
mark#( U13( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U14( X1 , X2 , X3 ) )	→	a__U14#( mark( X1 ) , X2 , X3 )
mark#( U14( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U15( X1 , X2 ) )	→	a__U15#( mark( X1 ) , X2 )
mark#( U15( X1 , X2 ) )	→	mark#( X1 )
mark#( isNat( X ) )	→	a__isNat#( X )
mark#( U16( X ) )	→	a__U16#( mark( X ) )
mark#( U16( X ) )	→	mark#( X )
mark#( U21( X1 , X2 ) )	→	a__U21#( mark( X1 ) , X2 )
mark#( U21( X1 , X2 ) )	→	mark#( X1 )
mark#( U22( X1 , X2 ) )	→	a__U22#( mark( X1 ) , X2 )
mark#( U22( X1 , X2 ) )	→	mark#( X1 )
mark#( U23( X ) )	→	a__U23#( mark( X ) )
mark#( U23( X ) )	→	mark#( X )
mark#( U31( X1 , X2 ) )	→	a__U31#( mark( X1 ) , X2 )
mark#( U31( X1 , X2 ) )	→	mark#( X1 )
mark#( U32( X ) )	→	a__U32#( mark( X ) )
mark#( U32( X ) )	→	mark#( X )
mark#( U41( X ) )	→	a__U41#( mark( X ) )
mark#( U41( X ) )	→	mark#( X )
mark#( U51( X1 , X2 ) )	→	a__U51#( mark( X1 ) , X2 )
mark#( U51( X1 , X2 ) )	→	mark#( X1 )
mark#( U52( X1 , X2 ) )	→	a__U52#( mark( X1 ) , X2 )
mark#( U52( X1 , X2 ) )	→	mark#( X1 )
mark#( U61( X1 , X2 , X3 ) )	→	a__U61#( mark( X1 ) , X2 , X3 )
mark#( U61( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U62( X1 , X2 , X3 ) )	→	a__U62#( mark( X1 ) , X2 , X3 )
mark#( U62( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U63( X1 , X2 , X3 ) )	→	a__U63#( mark( X1 ) , X2 , X3 )
mark#( U63( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U64( X1 , X2 , X3 ) )	→	a__U64#( mark( X1 ) , X2 , X3 )
mark#( U64( 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#( s( X ) )	→	mark#( X )

[isNatKind (x₁) ]	=	0
[a__U51 (x₁, x₂) ]	=	x₁ + x₂ + 1
[a__U16 (x₁) ]	=	3 x₁
[a__plus (x₁, x₂) ]	=	2 x₁ + x₂
[0]	=	1
[U21 (x₁, x₂) ]	=	x₁
[U31 (x₁, x₂) ]	=	x₁
[a__U23 (x₁) ]	=	2 x₁
[a__U12 (x₁, x₂, x₃) ]	=	2 x₁
[a__U14 (x₁, x₂, x₃) ]	=	x₁
[U12 (x₁, x₂, x₃) ]	=	2 x₁
[a__U52# (x₁, x₂) ]	=	2 x₁
[a__U51# (x₁, x₂) ]	=	2 x₁ + 2
[a__U21 (x₁, x₂) ]	=	x₁
[mark# (x₁) ]	=	2 x₁
[a__U13 (x₁, x₂, x₃) ]	=	x₁
[U63 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃ + 1
[a__U41 (x₁) ]	=	3 x₁
[tt]	=	0
[a__U61 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃ + 1
[a__U22 (x₁, x₂) ]	=	2 x₁
[a__U32 (x₁) ]	=	x₁
[U41 (x₁) ]	=	3 x₁
[a__isNatKind (x₁) ]	=	0
[a__U64# (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂
[plus (x₁, x₂) ]	=	2 x₁ + x₂
[U23 (x₁) ]	=	2 x₁
[a__isNat (x₁) ]	=	0
[a__U62# (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 2
[a__plus# (x₁, x₂) ]	=	2 x₁ + 2 x₂
[a__U63 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃ + 1
[U64 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃ + 1
[U11 (x₁, x₂, x₃) ]	=	x₁
[a__U63# (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 1
[a__U64 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃ + 1
[U61 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃ + 1
[U15 (x₁, x₂) ]	=	x₁
[a__U31 (x₁, x₂) ]	=	x₁
[U22 (x₁, x₂) ]	=	2 x₁
[a__U15 (x₁, x₂) ]	=	x₁
[U32 (x₁) ]	=	x₁
[s (x₁) ]	=	x₁ + 1
[U14 (x₁, x₂, x₃) ]	=	x₁
[a__U61# (x₁, x₂, x₃) ]	=	2 x₁ + 2 x₂ + 2
[mark (x₁) ]	=	x₁
[U13 (x₁, x₂, x₃) ]	=	x₁
[a__U52 (x₁, x₂) ]	=	2 x₁ + x₂
[U52 (x₁, x₂) ]	=	2 x₁ + x₂
[U51 (x₁, x₂) ]	=	x₁ + x₂ + 1
[a__U62 (x₁, x₂, x₃) ]	=	x₁ + x₂ + 2 x₃ + 1
[U16 (x₁) ]	=	3 x₁
[isNat (x₁) ]	=	0
[a__U11 (x₁, x₂, x₃) ]	=	x₁
[U62 (x₁, x₂, x₃) ]	=	x₁ + x₂ + 2 x₃ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U11 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 3 x₃ + 1
[isNatKind (x₁) ]	=	0
[a__U51 (x₁, x₂) ]	=	x₁
[a__U64 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[U61 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[a__U16 (x₁) ]	=	2 x₁
[a__plus (x₁, x₂) ]	=	x₁ + 3 x₂ + 1
[0]	=	0
[U15 (x₁, x₂) ]	=	x₁ + 2 x₂
[U21 (x₁, x₂) ]	=	x₁ + x₂ + 2
[a__U31 (x₁, x₂) ]	=	x₁
[U31 (x₁, x₂) ]	=	x₁
[U22 (x₁, x₂) ]	=	x₁ + x₂ + 2
[a__U23 (x₁) ]	=	x₁ + 2
[a__U12 (x₁, x₂, x₃) ]	=	x₁ + x₂ + 3 x₃
[a__U15 (x₁, x₂) ]	=	x₁ + 2 x₂
[U32 (x₁) ]	=	x₁
[a__U14 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃
[s (x₁) ]	=	x₁ + 2
[U12 (x₁, x₂, x₃) ]	=	x₁ + x₂ + 3 x₃
[U14 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃
[mark (x₁) ]	=	x₁
[U13 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃
[a__U52 (x₁, x₂) ]	=	x₁
[a__U21 (x₁, x₂) ]	=	x₁ + x₂ + 2
[U52 (x₁, x₂) ]	=	x₁
[mark# (x₁) ]	=	2 x₁
[a__U13 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 2 x₃
[U51 (x₁, x₂) ]	=	x₁
[U63 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[tt]	=	0
[a__U41 (x₁) ]	=	2 x₁
[a__U62 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[a__U61 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[U16 (x₁) ]	=	2 x₁
[isNat (x₁) ]	=	x₁
[a__U22 (x₁, x₂) ]	=	x₁ + x₂ + 2
[a__U11 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 3 x₃ + 1
[a__isNatKind (x₁) ]	=	0
[U41 (x₁) ]	=	2 x₁
[a__U32 (x₁) ]	=	x₁
[U62 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[plus (x₁, x₂) ]	=	x₁ + 3 x₂ + 1
[U23 (x₁) ]	=	x₁ + 2
[a__isNat (x₁) ]	=	x₁
[a__U63 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[U64 (x₁, x₂, x₃) ]	=	3 x₁ + x₂ + 3
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U11 (x₁, x₂, x₃) ]	=	x₁ + 2 x₂ + 1
[isNatKind (x₁) ]	=	x₁
[a__U64 (x₁, x₂, x₃) ]	=	2 x₁ + 2
[a__U51 (x₁, x₂) ]	=	2 x₁ + 2
[U61 (x₁, x₂, x₃) ]	=	x₁ + 2
[a__U16 (x₁) ]	=	0
[a__plus (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 2
[0]	=	0
[U15 (x₁, x₂) ]	=	0
[a__U31 (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 1
[U21 (x₁, x₂) ]	=	1
[U31 (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 1
[U22 (x₁, x₂) ]	=	1
[a__U23 (x₁) ]	=	1
[a__U15 (x₁, x₂) ]	=	0
[a__U12 (x₁, x₂, x₃) ]	=	2
[a__U14 (x₁, x₂, x₃) ]	=	2
[U32 (x₁) ]	=	x₁ + 1
[U12 (x₁, x₂, x₃) ]	=	2
[s (x₁) ]	=	2
[U14 (x₁, x₂, x₃) ]	=	1
[mark (x₁) ]	=	2 x₁
[U13 (x₁, x₂, x₃) ]	=	1
[a__U21 (x₁, x₂) ]	=	2
[a__U52 (x₁, x₂) ]	=	2 x₁
[U52 (x₁, x₂) ]	=	x₁
[mark# (x₁) ]	=	2 x₁
[a__U13 (x₁, x₂, x₃) ]	=	2
[U51 (x₁, x₂) ]	=	x₁ + 1
[U63 (x₁, x₂, x₃) ]	=	2 x₁ + 2
[a__U62 (x₁, x₂, x₃) ]	=	2 x₁ + 3
[a__U41 (x₁) ]	=	0
[tt]	=	0
[a__U61 (x₁, x₂, x₃) ]	=	2 x₁ + 3
[isNat (x₁) ]	=	2 x₁ + 2
[U16 (x₁) ]	=	0
[a__U11 (x₁, x₂, x₃) ]	=	x₁ + 3 x₂ + 2
[a__U22 (x₁, x₂) ]	=	2
[a__U32 (x₁) ]	=	x₁ + 1
[a__isNatKind (x₁) ]	=	2 x₁
[U41 (x₁) ]	=	0
[U62 (x₁, x₂, x₃) ]	=	2 x₁ + 2
[plus (x₁, x₂) ]	=	2 x₁ + 2 x₂ + 1
[U23 (x₁) ]	=	1
[a__isNat (x₁) ]	=	3 x₁ + 2
[a__U63 (x₁, x₂, x₃) ]	=	2 x₁ + 3
[U64 (x₁, x₂, x₃) ]	=	x₁ + 2
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n