Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

a__U41#( tt , V2 ) → a__U42#( a__isNatIList( V2 ) )

a__U41#( tt , V2 ) → a__isNatIList#( V2 )

a__U51#( tt , V2 ) → a__U52#( a__isNatList( V2 ) )

a__U51#( tt , V2 ) → a__isNatList#( V2 )

a__U61#( tt , L , N ) → a__U62#( a__isNat( N ) , L )

a__U61#( tt , L , N ) → a__isNat#( N )

a__U62#( tt , L ) → a__length#( mark( L ) )

a__U62#( tt , L ) → mark#( L )

a__isNat#( length( V1 ) ) → a__U11#( a__isNatList( V1 ) )

a__isNat#( length( V1 ) ) → a__isNatList#( V1 )

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

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

a__isNatIList#( V ) → a__U31#( a__isNatList( V ) )

a__isNatIList#( V ) → a__isNatList#( V )

a__isNatIList#( cons( V1 , V2 ) ) → a__U41#( a__isNat( V1 ) , V2 )

a__isNatIList#( cons( V1 , V2 ) ) → a__isNat#( V1 )

a__isNatList#( cons( V1 , V2 ) ) → a__U51#( a__isNat( V1 ) , V2 )

a__isNatList#( cons( V1 , V2 ) ) → a__isNat#( V1 )

a__length#( cons( N , L ) ) → a__U61#( a__isNatList( L ) , L , N )

a__length#( cons( N , L ) ) → a__isNatList#( L )

mark#( zeros ) → a__zeros#

mark#( U11( X ) ) → a__U11#( mark( X ) )

mark#( U11( X ) ) → mark#( X )

mark#( U21( X ) ) → a__U21#( mark( X ) )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X ) ) → a__U31#( mark( X ) )

mark#( U31( X ) ) → mark#( X )

mark#( U41( X1 , X2 ) ) → a__U41#( mark( X1 ) , X2 )

mark#( U41( X1 , X2 ) ) → mark#( X1 )

mark#( U42( X ) ) → a__U42#( mark( X ) )

mark#( U42( X ) ) → mark#( X )

mark#( isNatIList( X ) ) → a__isNatIList#( X )

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

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → a__U52#( mark( X ) )

mark#( U52( X ) ) → mark#( X )

mark#( isNatList( X ) ) → a__isNatList#( X )

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

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

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

mark#( U62( X1 , X2 ) ) → mark#( X1 )

mark#( isNat( X ) ) → a__isNat#( X )

mark#( length( X ) ) → a__length#( mark( X ) )

mark#( length( X ) ) → mark#( X )

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

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( X ) ) → mark#( X )

mark#( U21( X ) ) → mark#( X )

mark#( U31( X ) ) → mark#( X )

mark#( U41( X1 , X2 ) ) → mark#( X1 )

mark#( U42( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

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

a__U61#( tt , L , N ) → a__U62#( a__isNat( N ) , L )

a__U62#( tt , L ) → a__length#( mark( L ) )

a__length#( cons( N , L ) ) → a__U61#( a__isNatList( L ) , L , N )

a__U62#( tt , L ) → mark#( L )

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

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

mark#( U62( X1 , X2 ) ) → mark#( X1 )

mark#( length( X ) ) → a__length#( mark( X ) )

mark#( length( X ) ) → mark#( X )

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

mark#( s( X ) ) → mark#( X )

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = x₁ + 1

[length (x₁) ] = 3 x₁

[a__length (x₁) ] = 3 x₁

[a__isNatIList (x₁) ] = 1

[a__U51 (x₁, x₂) ] = 2 x₁

[U61 (x₁, x₂, x₃) ] = x₁ + 3 x₂ + x₃

[0] = 0

[cons (x₁, x₂) ] = x₁ + 2 x₂

[a__length# (x₁) ] = 2 x₁

[isNatList (x₁) ] = 0

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = x₁ + 1

[a__U11 (x₁) ] = x₁

[U52 (x₁) ] = 2 x₁

[s (x₁) ] = x₁

[U11 (x₁) ] = x₁

[isNatIList (x₁) ] = 1

[a__U42 (x₁) ] = x₁

[a__U61# (x₁, x₂, x₃) ] = 3 x₁

[mark (x₁) ] = x₁

[zeros] = 0

[U62 (x₁, x₂) ] = x₁ + 3 x₂

[U41 (x₁, x₂) ] = x₁ + 1

[a__zeros] = 0

[U31 (x₁) ] = x₁ + 1

[mark# (x₁) ] = x₁

[U51 (x₁, x₂) ] = 2 x₁

[a__U62# (x₁, x₂) ] = 3 x₁

[nil] = 0

[a__isNatList (x₁) ] = 0

[tt] = 0

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

[isNat (x₁) ] = 0

[U21 (x₁) ] = x₁

[a__U52 (x₁) ] = 2 x₁

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = x₁ + 3 x₂

[U42 (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( X ) ) → mark#( X )

mark#( U21( X ) ) → mark#( X )

mark#( U42( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

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

a__U61#( tt , L , N ) → a__U62#( a__isNat( N ) , L )

a__U62#( tt , L ) → a__length#( mark( L ) )

a__length#( cons( N , L ) ) → a__U61#( a__isNatList( L ) , L , N )

a__U62#( tt , L ) → mark#( L )

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

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

mark#( U62( X1 , X2 ) ) → mark#( X1 )

mark#( length( X ) ) → a__length#( mark( X ) )

mark#( length( X ) ) → mark#( X )

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

mark#( s( X ) ) → mark#( X )

1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 0

[length (x₁) ] = x₁ + 1

[a__length (x₁) ] = x₁ + 1

[a__isNatIList (x₁) ] = 0

[a__U51 (x₁, x₂) ] = 2 x₁

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

[0] = 0

[cons (x₁, x₂) ] = 2 x₁ + 2 x₂

[a__length# (x₁) ] = x₁

[isNatList (x₁) ] = 0

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = 0

[a__U11 (x₁) ] = x₁

[U52 (x₁) ] = 2 x₁

[s (x₁) ] = x₁

[U11 (x₁) ] = x₁

[isNatIList (x₁) ] = 0

[a__U42 (x₁) ] = x₁

[a__U61# (x₁, x₂, x₃) ] = x₁ + 2 x₂

[mark (x₁) ] = x₁

[zeros] = 0

[U62 (x₁, x₂) ] = 2 x₁ + x₂ + 1

[U41 (x₁, x₂) ] = 0

[a__zeros] = 0

[U31 (x₁) ] = 0

[mark# (x₁) ] = x₁

[U51 (x₁, x₂) ] = 2 x₁

[a__U62# (x₁, x₂) ] = x₁

[nil] = 0

[a__isNatList (x₁) ] = 0

[tt] = 0

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

[isNat (x₁) ] = 0

[U21 (x₁) ] = x₁

[a__U52 (x₁) ] = 2 x₁

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = 2 x₁ + x₂ + 1

[U42 (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( X ) ) → mark#( X )

mark#( U21( X ) ) → mark#( X )

mark#( U42( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

a__U61#( tt , L , N ) → a__U62#( a__isNat( N ) , L )

a__U62#( tt , L ) → a__length#( mark( L ) )

a__length#( cons( N , L ) ) → a__U61#( a__isNatList( L ) , L , N )

a__U62#( tt , L ) → mark#( L )

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

mark#( s( X ) ) → mark#( X )

1.1.1.1.1: dependency graph processor

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

The 1st component contains the pair(s)

a__U62#( tt , L ) → a__length#( mark( L ) )

a__length#( cons( N , L ) ) → a__U61#( a__isNatList( L ) , L , N )

a__U61#( tt , L , N ) → a__U62#( a__isNat( N ) , L )

1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 1

[length (x₁) ] = 2 x₁

[a__length (x₁) ] = 2 x₁

[a__isNatIList (x₁) ] = 1

[a__U51 (x₁, x₂) ] = 3 x₁

[U61 (x₁, x₂, x₃) ] = 2 x₁

[0] = 2

[a__length# (x₁) ] = 2 x₁

[cons (x₁, x₂) ] = 3 x₁

[isNatList (x₁) ] = 2 x₁

[a__U21 (x₁) ] = 1

[a__U41 (x₁, x₂) ] = 1

[a__U11 (x₁) ] = x₁

[U52 (x₁) ] = x₁

[s (x₁) ] = 1

[U11 (x₁) ] = x₁

[isNatIList (x₁) ] = 1

[a__U42 (x₁) ] = x₁

[a__U61# (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[zeros] = 0

[mark (x₁) ] = x₁

[U62 (x₁, x₂) ] = 1

[U41 (x₁, x₂) ] = 1

[a__zeros] = 0

[U31 (x₁) ] = 1

[U51 (x₁, x₂) ] = 3 x₁

[a__U62# (x₁, x₂) ] = 2 x₁

[nil] = 1

[a__isNatList (x₁) ] = 2 x₁

[tt] = 1

[a__U61 (x₁, x₂, x₃) ] = 2 x₁

[isNat (x₁) ] = 2 x₁

[a__U52 (x₁) ] = x₁

[U21 (x₁) ] = 1

[a__isNat (x₁) ] = 2 x₁

[a__U62 (x₁, x₂) ] = 1

[U42 (x₁) ] = x₁

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

one remains with the following pair(s).

a__U62#( tt , L ) → a__length#( mark( L ) )

a__length#( cons( N , L ) ) → a__U61#( a__isNatList( L ) , L , N )

1.1.1.1.1.1.1: dependency graph processor

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

The 2nd component contains the pair(s)

mark#( U21( X ) ) → mark#( X )

mark#( U11( X ) ) → mark#( X )

mark#( U42( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

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

mark#( s( X ) ) → mark#( X )

1.1.1.1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 0

[length (x₁) ] = 0

[a__length (x₁) ] = 0

[a__isNatIList (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁

[U61 (x₁, x₂, x₃) ] = 0

[0] = 0

[cons (x₁, x₂) ] = x₁ + 1

[isNatList (x₁) ] = 0

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = 2 x₁

[a__U11 (x₁) ] = 2 x₁

[U52 (x₁) ] = 2 x₁

[s (x₁) ] = 2 x₁

[U11 (x₁) ] = 2 x₁

[isNatIList (x₁) ] = 0

[a__U42 (x₁) ] = 2 x₁

[mark (x₁) ] = 2 x₁

[zeros] = 1

[U62 (x₁, x₂) ] = 0

[U41 (x₁, x₂) ] = 2 x₁

[a__zeros] = 2

[U31 (x₁) ] = 0

[mark# (x₁) ] = x₁

[U51 (x₁, x₂) ] = x₁

[nil] = 2

[a__isNatList (x₁) ] = 0

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 0

[isNat (x₁) ] = 0

[a__U52 (x₁) ] = 2 x₁

[U21 (x₁) ] = x₁

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = 0

[U42 (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#( U21( X ) ) → mark#( X )

mark#( U11( X ) ) → mark#( X )

mark#( U42( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

mark#( s( X ) ) → mark#( X )

1.1.1.1.1.2.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 0

[length (x₁) ] = 0

[a__length (x₁) ] = 0

[a__isNatIList (x₁) ] = 2 x₁

[a__U51 (x₁, x₂) ] = 2 x₁

[U61 (x₁, x₂, x₃) ] = 0

[0] = 0

[cons (x₁, x₂) ] = 2 x₁ + x₂ + 2

[isNatList (x₁) ] = 0

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = 2 x₁ + 3

[a__U11 (x₁) ] = 2 x₁

[U52 (x₁) ] = 2 x₁

[s (x₁) ] = 2 x₁

[U11 (x₁) ] = 2 x₁

[isNatIList (x₁) ] = 2 x₁

[a__U42 (x₁) ] = x₁ + 2

[mark (x₁) ] = 3 x₁

[zeros] = 1

[U62 (x₁, x₂) ] = 0

[U41 (x₁, x₂) ] = 2 x₁ + 3

[a__zeros] = 3

[U31 (x₁) ] = 0

[mark# (x₁) ] = 2 x₁

[U51 (x₁, x₂) ] = 2 x₁

[nil] = 3

[a__isNatList (x₁) ] = 0

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 0

[isNat (x₁) ] = 0

[a__U52 (x₁) ] = 2 x₁

[U21 (x₁) ] = x₁

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = 0

[U42 (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#( U21( X ) ) → mark#( X )

mark#( U11( X ) ) → mark#( X )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

mark#( s( X ) ) → mark#( X )

1.1.1.1.1.2.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 0

[length (x₁) ] = x₁ + 1

[a__length (x₁) ] = x₁ + 1

[a__isNatIList (x₁) ] = 0

[a__U51 (x₁, x₂) ] = x₁ + 2 x₂

[U61 (x₁, x₂, x₃) ] = x₁ + 1

[0] = 0

[cons (x₁, x₂) ] = x₁ + x₂

[isNatList (x₁) ] = 2 x₁

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = 0

[a__U11 (x₁) ] = x₁ + 1

[U52 (x₁) ] = x₁

[s (x₁) ] = x₁

[U11 (x₁) ] = x₁ + 1

[isNatIList (x₁) ] = 0

[a__U42 (x₁) ] = 0

[zeros] = 0

[mark (x₁) ] = x₁

[U62 (x₁, x₂) ] = x₁ + 1

[U41 (x₁, x₂) ] = 0

[a__zeros] = 0

[U31 (x₁) ] = 0

[mark# (x₁) ] = 2 x₁

[U51 (x₁, x₂) ] = x₁ + 2 x₂

[nil] = 0

[a__isNatList (x₁) ] = 2 x₁

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = x₁ + 1

[isNat (x₁) ] = 2 x₁

[a__U52 (x₁) ] = x₁

[U21 (x₁) ] = x₁

[a__isNat (x₁) ] = 2 x₁

[a__U62 (x₁, x₂) ] = x₁ + 1

[U42 (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 )

mark#( U51( X1 , X2 ) ) → mark#( X1 )

mark#( U52( X ) ) → mark#( X )

mark#( s( X ) ) → mark#( X )

1.1.1.1.1.2.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 2 x₁ + 1

[length (x₁) ] = 0

[a__length (x₁) ] = 0

[a__isNatIList (x₁) ] = 2 x₁ + 3

[a__U51 (x₁, x₂) ] = 2 x₁ + x₂ + 1

[U61 (x₁, x₂, x₃) ] = 0

[0] = 0

[cons (x₁, x₂) ] = 2 x₁

[isNatList (x₁) ] = x₁ + 1

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = 2 x₁

[a__U11 (x₁) ] = 0

[U52 (x₁) ] = x₁

[s (x₁) ] = x₁

[U11 (x₁) ] = 0

[isNatIList (x₁) ] = x₁ + 2

[a__U42 (x₁) ] = 0

[zeros] = 0

[mark (x₁) ] = 2 x₁

[U62 (x₁, x₂) ] = 0

[U41 (x₁, x₂) ] = x₁

[a__zeros] = 0

[U31 (x₁) ] = 2 x₁ + 1

[mark# (x₁) ] = x₁

[U51 (x₁, x₂) ] = 2 x₁ + x₂ + 1

[nil] = 2

[a__isNatList (x₁) ] = x₁ + 1

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 0

[isNat (x₁) ] = 0

[a__U52 (x₁) ] = x₁

[U21 (x₁) ] = x₁

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = 0

[U42 (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 )

mark#( U52( X ) ) → mark#( X )

mark#( s( X ) ) → mark#( X )

1.1.1.1.1.2.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 3

[length (x₁) ] = 2 x₁

[a__length (x₁) ] = 2 x₁

[a__isNatIList (x₁) ] = 3

[a__U51 (x₁, x₂) ] = 2 x₁

[U61 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[0] = 0

[cons (x₁, x₂) ] = 2 x₁

[isNatList (x₁) ] = x₁

[a__U21 (x₁) ] = x₁

[a__U41 (x₁, x₂) ] = x₁ + 1

[a__U11 (x₁) ] = 2

[U52 (x₁) ] = 2 x₁

[s (x₁) ] = x₁ + 2

[U11 (x₁) ] = 2

[isNatIList (x₁) ] = 3

[a__U42 (x₁) ] = x₁

[zeros] = 0

[mark (x₁) ] = x₁

[U62 (x₁, x₂) ] = 2 x₁ + 2 x₂

[U41 (x₁, x₂) ] = x₁ + 1

[a__zeros] = 0

[U31 (x₁) ] = 3

[mark# (x₁) ] = x₁

[U51 (x₁, x₂) ] = 2 x₁

[nil] = 3

[a__isNatList (x₁) ] = x₁

[tt] = 2

[a__U61 (x₁, x₂, x₃) ] = 2 x₁ + 2 x₂

[isNat (x₁) ] = 2

[a__U52 (x₁) ] = 2 x₁

[U21 (x₁) ] = x₁

[a__isNat (x₁) ] = 2

[a__U62 (x₁, x₂) ] = 2 x₁ + 2 x₂

[U42 (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#( U21( X ) ) → mark#( X )

mark#( U52( X ) ) → mark#( X )

1.1.1.1.1.2.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 0

[length (x₁) ] = 0

[a__length (x₁) ] = 0

[a__isNatIList (x₁) ] = 3 x₁ + 1

[a__U51 (x₁, x₂) ] = 2 x₁ + 2

[U61 (x₁, x₂, x₃) ] = 0

[0] = 0

[cons (x₁, x₂) ] = x₁ + 1

[isNatList (x₁) ] = 2 x₁

[a__U21 (x₁) ] = 2 x₁

[a__U41 (x₁, x₂) ] = 3 x₁ + 3

[a__U11 (x₁) ] = 0

[U52 (x₁) ] = x₁ + 1

[s (x₁) ] = 0

[U11 (x₁) ] = 0

[isNatIList (x₁) ] = 2 x₁ + 1

[a__U42 (x₁) ] = 1

[zeros] = 1

[mark (x₁) ] = 2 x₁

[U62 (x₁, x₂) ] = 0

[U41 (x₁, x₂) ] = 3 x₁ + 2

[a__zeros] = 2

[U31 (x₁) ] = 0

[mark# (x₁) ] = 3 x₁

[U51 (x₁, x₂) ] = 2 x₁ + 1

[nil] = 0

[a__isNatList (x₁) ] = 2 x₁

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 0

[isNat (x₁) ] = 0

[a__U52 (x₁) ] = x₁ + 1

[U21 (x₁) ] = 2 x₁

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = 0

[U42 (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#( U21( X ) ) → mark#( X )

1.1.1.1.1.2.1.1.1.1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 1

[length (x₁) ] = x₁

[a__length (x₁) ] = x₁

[a__isNatIList (x₁) ] = 2

[a__U51 (x₁, x₂) ] = 2 x₁

[U61 (x₁, x₂, x₃) ] = x₁ + 2 x₂

[0] = 1

[cons (x₁, x₂) ] = 3 x₁

[isNatList (x₁) ] = x₁

[a__U21 (x₁) ] = 2 x₁ + 1

[a__U41 (x₁, x₂) ] = 2

[a__U11 (x₁) ] = x₁

[U52 (x₁) ] = x₁

[s (x₁) ] = 2 x₁ + 1

[U11 (x₁) ] = x₁

[isNatIList (x₁) ] = 2

[a__U42 (x₁) ] = x₁

[zeros] = 0

[mark (x₁) ] = x₁

[U62 (x₁, x₂) ] = 2 x₁ + 1

[U41 (x₁, x₂) ] = 2

[a__zeros] = 0

[U31 (x₁) ] = 1

[mark# (x₁) ] = 2 x₁

[U51 (x₁, x₂) ] = 2 x₁

[nil] = 2

[a__isNatList (x₁) ] = x₁

[tt] = 1

[a__U61 (x₁, x₂, x₃) ] = x₁ + 2 x₂

[isNat (x₁) ] = 2 x₁

[a__U52 (x₁) ] = x₁

[U21 (x₁) ] = 2 x₁ + 1

[a__isNat (x₁) ] = 2 x₁

[a__U62 (x₁, x₂) ] = 2 x₁ + 1

[U42 (x₁) ] = x₁

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

one remains with the following pair(s).

none

1.1.1.1.1.2.1.1.1.1.1.1.1: P is empty

All dependency pairs have been removed.

The 2nd component contains the pair(s)

a__U41#( tt , V2 ) → a__isNatIList#( V2 )

a__isNatIList#( cons( V1 , V2 ) ) → a__U41#( a__isNat( V1 ) , V2 )

1.1.2: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 1

[length (x₁) ] = 0

[a__length (x₁) ] = 0

[a__isNatIList (x₁) ] = 1

[a__U51 (x₁, x₂) ] = 3 x₁ + 1

[U61 (x₁, x₂, x₃) ] = 0

[a__U41# (x₁, x₂) ] = 2 x₁

[0] = 0

[cons (x₁, x₂) ] = 2 x₁ + 2

[isNatList (x₁) ] = 2 x₁

[a__U21 (x₁) ] = 0

[a__U41 (x₁, x₂) ] = 0

[a__U11 (x₁) ] = 0

[U52 (x₁) ] = 0

[s (x₁) ] = 0

[U11 (x₁) ] = 0

[isNatIList (x₁) ] = 1

[a__U42 (x₁) ] = 0

[zeros] = 0

[mark (x₁) ] = 2 x₁ + 2

[U62 (x₁, x₂) ] = x₁

[U41 (x₁, x₂) ] = 0

[a__zeros] = 2

[U31 (x₁) ] = 1

[U51 (x₁, x₂) ] = 2 x₁ + 1

[nil] = 2

[a__isNatIList# (x₁) ] = 2 x₁

[a__isNatList (x₁) ] = 2 x₁

[tt] = 0

[a__U61 (x₁, x₂, x₃) ] = 0

[isNat (x₁) ] = 0

[a__U52 (x₁) ] = 0

[U21 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = x₁

[U42 (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).

a__U41#( tt , V2 ) → a__isNatIList#( V2 )

1.1.2.1: dependency graph processor

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

The 3rd component contains the pair(s)

a__U51#( tt , V2 ) → a__isNatList#( V2 )

a__isNatList#( cons( V1 , V2 ) ) → a__U51#( a__isNat( V1 ) , V2 )

a__isNatList#( cons( V1 , V2 ) ) → a__isNat#( V1 )

a__isNat#( length( V1 ) ) → a__isNatList#( V1 )

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

1.1.3: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 2

[a__isNatList# (x₁) ] = 2 x₁

[length (x₁) ] = x₁

[a__length (x₁) ] = x₁

[a__isNatIList (x₁) ] = x₁ + 2

[a__U51 (x₁, x₂) ] = 0

[U61 (x₁, x₂, x₃) ] = x₁ + x₂

[0] = 0

[cons (x₁, x₂) ] = x₁ + 2 x₂ + 1

[isNatList (x₁) ] = 0

[a__U21 (x₁) ] = 0

[a__U41 (x₁, x₂) ] = 2 x₁

[a__U11 (x₁) ] = 0

[U52 (x₁) ] = 0

[s (x₁) ] = x₁

[U11 (x₁) ] = 0

[isNatIList (x₁) ] = x₁ + 1

[a__U51# (x₁, x₂) ] = 3 x₁

[a__U42 (x₁) ] = 0

[zeros] = 0

[mark (x₁) ] = 2 x₁ + 1

[U62 (x₁, x₂) ] = x₁

[U41 (x₁, x₂) ] = x₁

[a__zeros] = 1

[U31 (x₁) ] = 2

[U51 (x₁, x₂) ] = 0

[nil] = 0

[a__isNatList (x₁) ] = 0

[tt] = 0

[a__isNat# (x₁) ] = 2 x₁ + 1

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

[isNat (x₁) ] = 0

[a__U52 (x₁) ] = 0

[U21 (x₁) ] = 0

[a__isNat (x₁) ] = 0

[a__U62 (x₁, x₂) ] = 2 x₁ + 1

[U42 (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).

a__U51#( tt , V2 ) → a__isNatList#( V2 )

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

1.1.3.1: dependency graph processor

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

The 1st component contains the pair(s)

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

1.1.3.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[a__U31 (x₁) ] = 2

[length (x₁) ] = x₁

[a__length (x₁) ] = x₁

[a__isNatIList (x₁) ] = 2

[a__U51 (x₁, x₂) ] = x₁

[U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂

[0] = 0

[cons (x₁, x₂) ] = 3 x₁

[isNatList (x₁) ] = x₁

[a__U21 (x₁) ] = 3

[a__U41 (x₁, x₂) ] = 2

[a__U11 (x₁) ] = x₁

[U52 (x₁) ] = x₁

[s (x₁) ] = x₁ + 1

[U11 (x₁) ] = x₁

[isNatIList (x₁) ] = 2

[a__U42 (x₁) ] = 2

[zeros] = 0

[mark (x₁) ] = x₁

[U62 (x₁, x₂) ] = x₁ + 1

[U41 (x₁, x₂) ] = 2

[a__zeros] = 0

[U31 (x₁) ] = 2

[U51 (x₁, x₂) ] = x₁

[nil] = 2

[a__isNatList (x₁) ] = x₁

[a__isNat# (x₁) ] = x₁

[tt] = 2

[a__U61 (x₁, x₂, x₃) ] = 2 x₁ + x₂

[isNat (x₁) ] = 2 x₁ + 2

[U21 (x₁) ] = 3

[a__U52 (x₁) ] = x₁

[a__isNat (x₁) ] = 2 x₁ + 2

[a__U62 (x₁, x₂) ] = x₁ + 1

[U42 (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

1.1.3.1.1.1: P is empty

All dependency pairs have been removed.

a__U41#( tt , V2 )	→	a__U42#( a__isNatIList( V2 ) )
a__U41#( tt , V2 )	→	a__isNatIList#( V2 )
a__U51#( tt , V2 )	→	a__U52#( a__isNatList( V2 ) )
a__U51#( tt , V2 )	→	a__isNatList#( V2 )
a__U61#( tt , L , N )	→	a__U62#( a__isNat( N ) , L )
a__U61#( tt , L , N )	→	a__isNat#( N )
a__U62#( tt , L )	→	a__length#( mark( L ) )
a__U62#( tt , L )	→	mark#( L )
a__isNat#( length( V1 ) )	→	a__U11#( a__isNatList( V1 ) )
a__isNat#( length( V1 ) )	→	a__isNatList#( V1 )
a__isNat#( s( V1 ) )	→	a__U21#( a__isNat( V1 ) )
a__isNat#( s( V1 ) )	→	a__isNat#( V1 )
a__isNatIList#( V )	→	a__U31#( a__isNatList( V ) )
a__isNatIList#( V )	→	a__isNatList#( V )
a__isNatIList#( cons( V1 , V2 ) )	→	a__U41#( a__isNat( V1 ) , V2 )
a__isNatIList#( cons( V1 , V2 ) )	→	a__isNat#( V1 )
a__isNatList#( cons( V1 , V2 ) )	→	a__U51#( a__isNat( V1 ) , V2 )
a__isNatList#( cons( V1 , V2 ) )	→	a__isNat#( V1 )
a__length#( cons( N , L ) )	→	a__U61#( a__isNatList( L ) , L , N )
a__length#( cons( N , L ) )	→	a__isNatList#( L )
mark#( zeros )	→	a__zeros#
mark#( U11( X ) )	→	a__U11#( mark( X ) )
mark#( U11( X ) )	→	mark#( X )
mark#( U21( X ) )	→	a__U21#( mark( X ) )
mark#( U21( X ) )	→	mark#( X )
mark#( U31( X ) )	→	a__U31#( mark( X ) )
mark#( U31( X ) )	→	mark#( X )
mark#( U41( X1 , X2 ) )	→	a__U41#( mark( X1 ) , X2 )
mark#( U41( X1 , X2 ) )	→	mark#( X1 )
mark#( U42( X ) )	→	a__U42#( mark( X ) )
mark#( U42( X ) )	→	mark#( X )
mark#( isNatIList( X ) )	→	a__isNatIList#( X )
mark#( U51( X1 , X2 ) )	→	a__U51#( mark( X1 ) , X2 )
mark#( U51( X1 , X2 ) )	→	mark#( X1 )
mark#( U52( X ) )	→	a__U52#( mark( X ) )
mark#( U52( X ) )	→	mark#( X )
mark#( isNatList( X ) )	→	a__isNatList#( X )
mark#( U61( X1 , X2 , X3 ) )	→	a__U61#( mark( X1 ) , X2 , X3 )
mark#( U61( X1 , X2 , X3 ) )	→	mark#( X1 )
mark#( U62( X1 , X2 ) )	→	a__U62#( mark( X1 ) , X2 )
mark#( U62( X1 , X2 ) )	→	mark#( X1 )
mark#( isNat( X ) )	→	a__isNat#( X )
mark#( length( X ) )	→	a__length#( mark( X ) )
mark#( length( X ) )	→	mark#( X )
mark#( cons( X1 , X2 ) )	→	mark#( X1 )
mark#( s( X ) )	→	mark#( X )

[a__U31 (x₁) ]	=	x₁ + 1
[length (x₁) ]	=	3 x₁
[a__length (x₁) ]	=	3 x₁
[a__isNatIList (x₁) ]	=	1
[a__U51 (x₁, x₂) ]	=	2 x₁
[U61 (x₁, x₂, x₃) ]	=	x₁ + 3 x₂ + x₃
[0]	=	0
[cons (x₁, x₂) ]	=	x₁ + 2 x₂
[a__length# (x₁) ]	=	2 x₁
[isNatList (x₁) ]	=	0
[a__U21 (x₁) ]	=	x₁
[a__U41 (x₁, x₂) ]	=	x₁ + 1
[a__U11 (x₁) ]	=	x₁
[U52 (x₁) ]	=	2 x₁
[s (x₁) ]	=	x₁
[U11 (x₁) ]	=	x₁
[isNatIList (x₁) ]	=	1
[a__U42 (x₁) ]	=	x₁
[a__U61# (x₁, x₂, x₃) ]	=	3 x₁
[mark (x₁) ]	=	x₁
[zeros]	=	0
[U62 (x₁, x₂) ]	=	x₁ + 3 x₂
[U41 (x₁, x₂) ]	=	x₁ + 1
[a__zeros]	=	0
[U31 (x₁) ]	=	x₁ + 1
[mark# (x₁) ]	=	x₁
[U51 (x₁, x₂) ]	=	2 x₁
[a__U62# (x₁, x₂) ]	=	3 x₁
[nil]	=	0
[a__isNatList (x₁) ]	=	0
[tt]	=	0
[a__U61 (x₁, x₂, x₃) ]	=	x₁ + 3 x₂ + x₃
[isNat (x₁) ]	=	0
[U21 (x₁) ]	=	x₁
[a__U52 (x₁) ]	=	2 x₁
[a__isNat (x₁) ]	=	0
[a__U62 (x₁, x₂) ]	=	x₁ + 3 x₂
[U42 (x₁) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[a__U31 (x₁) ]	=	2 x₁ + 1
[length (x₁) ]	=	0
[a__length (x₁) ]	=	0
[a__isNatIList (x₁) ]	=	2 x₁ + 3
[a__U51 (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[U61 (x₁, x₂, x₃) ]	=	0
[0]	=	0
[cons (x₁, x₂) ]	=	2 x₁
[isNatList (x₁) ]	=	x₁ + 1
[a__U21 (x₁) ]	=	x₁
[a__U41 (x₁, x₂) ]	=	2 x₁
[a__U11 (x₁) ]	=	0
[U52 (x₁) ]	=	x₁
[s (x₁) ]	=	x₁
[U11 (x₁) ]	=	0
[isNatIList (x₁) ]	=	x₁ + 2
[a__U42 (x₁) ]	=	0
[zeros]	=	0
[mark (x₁) ]	=	2 x₁
[U62 (x₁, x₂) ]	=	0
[U41 (x₁, x₂) ]	=	x₁
[a__zeros]	=	0
[U31 (x₁) ]	=	2 x₁ + 1
[mark# (x₁) ]	=	x₁
[U51 (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[nil]	=	2
[a__isNatList (x₁) ]	=	x₁ + 1
[tt]	=	0
[a__U61 (x₁, x₂, x₃) ]	=	0
[isNat (x₁) ]	=	0
[a__U52 (x₁) ]	=	x₁
[U21 (x₁) ]	=	x₁
[a__isNat (x₁) ]	=	0
[a__U62 (x₁, x₂) ]	=	0
[U42 (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[a__U31 (x₁) ]	=	2
[a__isNatList# (x₁) ]	=	2 x₁
[length (x₁) ]	=	x₁
[a__length (x₁) ]	=	x₁
[a__isNatIList (x₁) ]	=	x₁ + 2
[a__U51 (x₁, x₂) ]	=	0
[U61 (x₁, x₂, x₃) ]	=	x₁ + x₂
[0]	=	0
[cons (x₁, x₂) ]	=	x₁ + 2 x₂ + 1
[isNatList (x₁) ]	=	0
[a__U21 (x₁) ]	=	0
[a__U41 (x₁, x₂) ]	=	2 x₁
[a__U11 (x₁) ]	=	0
[U52 (x₁) ]	=	0
[s (x₁) ]	=	x₁
[U11 (x₁) ]	=	0
[isNatIList (x₁) ]	=	x₁ + 1
[a__U51# (x₁, x₂) ]	=	3 x₁
[a__U42 (x₁) ]	=	0
[zeros]	=	0
[mark (x₁) ]	=	2 x₁ + 1
[U62 (x₁, x₂) ]	=	x₁
[U41 (x₁, x₂) ]	=	x₁
[a__zeros]	=	1
[U31 (x₁) ]	=	2
[U51 (x₁, x₂) ]	=	0
[nil]	=	0
[a__isNatList (x₁) ]	=	0
[tt]	=	0
[a__isNat# (x₁) ]	=	2 x₁ + 1
[a__U61 (x₁, x₂, x₃) ]	=	2 x₁ + x₂ + 1
[isNat (x₁) ]	=	0
[a__U52 (x₁) ]	=	0
[U21 (x₁) ]	=	0
[a__isNat (x₁) ]	=	0
[a__U62 (x₁, x₂) ]	=	2 x₁ + 1
[U42 (x₁) ]	=	0
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n