Termination proof

1: switching to dependency pairs

The following set of initial dependency pairs has been identified.

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

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

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

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

U12#( tt , V2 ) → U13#( isNat( activate( V2 ) ) )

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

U12#( tt , V2 ) → activate#( V2 )

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

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

U21#( tt , V1 ) → activate#( V1 )

U31#( tt , N ) → activate#( N )

U41#( tt , M , N ) → s#( plus( activate( N ) , activate( M ) ) )

U41#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

U41#( tt , M , N ) → activate#( N )

U41#( tt , M , N ) → activate#( M )

and#( tt , X ) → activate#( X )

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

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

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

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

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

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 ) ) → and#( isNatKind( activate( V1 ) ) , n__isNatKind( 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 ) ) → isNatKind#( activate( V1 ) )

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

plus#( N , 0 ) → U31#( and( isNat( N ) , n__isNatKind( N ) ) , N )

plus#( N , 0 ) → and#( isNat( N ) , n__isNatKind( N ) )

plus#( N , 0 ) → isNat#( N )

plus#( N , s( M ) ) → U41#( and( and( isNat( M ) , n__isNatKind( M ) ) , n__and( n__isNat( N ) , n__isNatKind( N ) ) ) , M , N )

plus#( N , s( M ) ) → and#( and( isNat( M ) , n__isNatKind( M ) ) , n__and( n__isNat( N ) , n__isNatKind( N ) ) )

plus#( N , s( M ) ) → and#( isNat( M ) , n__isNatKind( M ) )

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__isNatKind( X ) ) → isNatKind#( X )

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

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

activate#( n__and( X1 , X2 ) ) → and#( activate( X1 ) , X2 )

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

activate#( n__isNat( X ) ) → isNat#( X )

1.1: dependency graph processor

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

The 1st component contains the pair(s)

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

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

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

U12#( tt , V2 ) → activate#( V2 )

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

plus#( N , 0 ) → U31#( and( isNat( N ) , n__isNatKind( N ) ) , N )

U31#( tt , N ) → activate#( N )

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

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

activate#( n__isNatKind( X ) ) → isNatKind#( X )

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

and#( tt , X ) → activate#( X )

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

activate#( n__and( X1 , X2 ) ) → and#( activate( X1 ) , X2 )

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

activate#( n__isNat( X ) ) → isNat#( X )

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

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

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 ) ) → isNatKind#( activate( V1 ) )

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

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

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

U21#( tt , V1 ) → activate#( V1 )

plus#( N , 0 ) → and#( isNat( N ) , n__isNatKind( N ) )

plus#( N , 0 ) → isNat#( N )

plus#( N , s( M ) ) → U41#( and( and( isNat( M ) , n__isNatKind( M ) ) , n__and( n__isNat( N ) , n__isNatKind( N ) ) ) , M , N )

U41#( tt , M , N ) → plus#( activate( N ) , activate( M ) )

plus#( N , s( M ) ) → and#( and( isNat( M ) , n__isNatKind( M ) ) , n__and( n__isNat( N ) , n__isNatKind( N ) ) )

plus#( N , s( M ) ) → and#( isNat( M ) , n__isNatKind( M ) )

plus#( N , s( M ) ) → isNat#( M )

U41#( tt , M , N ) → activate#( N )

U41#( tt , M , N ) → activate#( M )

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

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

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

1.1.1: reduction pair processor

Using the following reduction pair

Linear polynomial interpretation over the naturals

[U11 (x₁, x₂, x₃) ] = x₁ + x₂ + x₃

[n__plus (x₁, x₂) ] = 3 x₁ + 3 x₂

[isNatKind (x₁) ] = x₁

[U22 (x₁) ] = 1

[n__0] = 0

[U13 (x₁) ] = 0

[U12 (x₁, x₂) ] = x₁ + x₂

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

[0] = 0

[U41 (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂ + 1

[isNat# (x₁) ] = x₁ + 1

[activate (x₁) ] = x₁

[U21 (x₁, x₂) ] = x₁ + 1

[isNatKind# (x₁) ] = x₁ + 1

[U31 (x₁, x₂) ] = x₁

[and (x₁, x₂) ] = x₁ + 2 x₂

[s (x₁) ] = x₁ + 1

[U31# (x₁, x₂) ] = x₁ + 1

[and# (x₁, x₂) ] = x₁ + 1

[U21# (x₁, x₂) ] = x₁ + 2

[n__isNatKind (x₁) ] = x₁

[U11# (x₁, x₂, x₃) ] = x₁ + x₂ + 1

[n__s (x₁) ] = x₁ + 1

[plus# (x₁, x₂) ] = 3 x₁ + 3 x₂ + 1

[U12# (x₁, x₂) ] = x₁ + 1

[tt] = 0

[isNat (x₁) ] = x₁

[activate# (x₁) ] = x₁ + 1

[plus (x₁, x₂) ] = 3 x₁ + 3 x₂

[n__and (x₁, x₂) ] = x₁ + 2 x₂

[n__isNat (x₁) ] = x₁

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

one remains with the following pair(s).

1.1.1.1: dependency graph processor

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

The 1st component contains the pair(s)

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

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

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

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

and#( tt , X ) → activate#( X )

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

plus#( N , 0 ) → U31#( and( isNat( N ) , n__isNatKind( N ) ) , N )

U31#( tt , N ) → activate#( N )

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

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

activate#( n__isNatKind( X ) ) → isNatKind#( X )

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

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

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

activate#( n__and( X1 , X2 ) ) → and#( activate( X1 ) , X2 )

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

activate#( n__isNat( X ) ) → isNat#( X )

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

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

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

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

plus#( N , 0 ) → and#( isNat( N ) , n__isNatKind( N ) )

plus#( N , 0 ) → isNat#( N )

U12#( tt , V2 ) → activate#( V2 )

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

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

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

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 + 3

[n__plus (x₁, x₂) ] = 2 x₁ + 3 x₂ + 3

[isNatKind (x₁) ] = x₁

[U22 (x₁) ] = 1

[n__0] = 2

[U13 (x₁) ] = 1

[U12 (x₁, x₂) ] = 1

[0] = 2

[U41 (x₁, x₂, x₃) ] = 3 x₁ + 2 x₂ + 3

[isNat# (x₁) ] = 2 x₁

[activate (x₁) ] = x₁

[U21 (x₁, x₂) ] = x₁ + 1

[isNatKind# (x₁) ] = 2 x₁

[and (x₁, x₂) ] = 2 x₁ + x₂ + 1

[U31 (x₁, x₂) ] = x₁ + 3

[s (x₁) ] = x₁

[and# (x₁, x₂) ] = 3 x₁ + 2

[U31# (x₁, x₂) ] = 3 x₁ + 1

[n__isNatKind (x₁) ] = x₁

[U11# (x₁, x₂, x₃) ] = 3 x₁ + 3 x₂

[n__s (x₁) ] = x₁

[plus# (x₁, x₂) ] = 3 x₁ + 2 x₂

[U12# (x₁, x₂) ] = 3 x₁

[tt] = 0

[isNat (x₁) ] = 2 x₁ + 1

[activate# (x₁) ] = 3 x₁

[plus (x₁, x₂) ] = 2 x₁ + 3 x₂ + 3

[n__and (x₁, x₂) ] = 2 x₁ + x₂ + 1

[n__isNat (x₁) ] = 2 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#( tt , V1 , V2 ) → U12#( isNat( activate( V1 ) ) , activate( V2 ) )

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

activate#( n__isNatKind( X ) ) → isNatKind#( X )

U12#( tt , V2 ) → activate#( V2 )

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

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

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

1.1.1.1.1.1: dependency graph processor

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

U11#( tt , V1 , V2 )	→	U12#( isNat( activate( V1 ) ) , activate( V2 ) )
U11#( tt , V1 , V2 )	→	isNat#( activate( V1 ) )
U11#( tt , V1 , V2 )	→	activate#( V1 )
U11#( tt , V1 , V2 )	→	activate#( V2 )
U12#( tt , V2 )	→	U13#( isNat( activate( V2 ) ) )
U12#( tt , V2 )	→	isNat#( activate( V2 ) )
U12#( tt , V2 )	→	activate#( V2 )
U21#( tt , V1 )	→	U22#( isNat( activate( V1 ) ) )
U21#( tt , V1 )	→	isNat#( activate( V1 ) )
U21#( tt , V1 )	→	activate#( V1 )
U31#( tt , N )	→	activate#( N )
U41#( tt , M , N )	→	s#( plus( activate( N ) , activate( M ) ) )
U41#( tt , M , N )	→	plus#( activate( N ) , activate( M ) )
U41#( tt , M , N )	→	activate#( N )
U41#( tt , M , N )	→	activate#( M )
and#( tt , X )	→	activate#( X )
isNat#( n__plus( V1 , V2 ) )	→	U11#( and( isNatKind( activate( V1 ) ) , n__isNatKind( activate( V2 ) ) ) , activate( V1 ) , activate( V2 ) )
isNat#( n__plus( V1 , V2 ) )	→	and#( isNatKind( activate( V1 ) ) , n__isNatKind( activate( V2 ) ) )
isNat#( n__plus( V1 , V2 ) )	→	isNatKind#( activate( V1 ) )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V1 )
isNat#( n__plus( V1 , V2 ) )	→	activate#( V2 )
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 ) )	→	and#( isNatKind( activate( V1 ) ) , n__isNatKind( 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 ) )	→	isNatKind#( activate( V1 ) )
isNatKind#( n__s( V1 ) )	→	activate#( V1 )
plus#( N , 0 )	→	U31#( and( isNat( N ) , n__isNatKind( N ) ) , N )
plus#( N , 0 )	→	and#( isNat( N ) , n__isNatKind( N ) )
plus#( N , 0 )	→	isNat#( N )
plus#( N , s( M ) )	→	U41#( and( and( isNat( M ) , n__isNatKind( M ) ) , n__and( n__isNat( N ) , n__isNatKind( N ) ) ) , M , N )
plus#( N , s( M ) )	→	and#( and( isNat( M ) , n__isNatKind( M ) ) , n__and( n__isNat( N ) , n__isNatKind( N ) ) )
plus#( N , s( M ) )	→	and#( isNat( M ) , n__isNatKind( M ) )
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__isNatKind( X ) )	→	isNatKind#( X )
activate#( n__s( X ) )	→	s#( activate( X ) )
activate#( n__s( X ) )	→	activate#( X )
activate#( n__and( X1 , X2 ) )	→	and#( activate( X1 ) , X2 )
activate#( n__and( X1 , X2 ) )	→	activate#( X1 )
activate#( n__isNat( X ) )	→	isNat#( X )

[U11 (x₁, x₂, x₃) ]	=	x₁ + x₂ + x₃
[n__plus (x₁, x₂) ]	=	3 x₁ + 3 x₂
[isNatKind (x₁) ]	=	x₁
[U22 (x₁) ]	=	1
[n__0]	=	0
[U13 (x₁) ]	=	0
[U12 (x₁, x₂) ]	=	x₁ + x₂
[U41# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂ + 2
[0]	=	0
[U41 (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂ + 1
[isNat# (x₁) ]	=	x₁ + 1
[activate (x₁) ]	=	x₁
[U21 (x₁, x₂) ]	=	x₁ + 1
[isNatKind# (x₁) ]	=	x₁ + 1
[U31 (x₁, x₂) ]	=	x₁
[and (x₁, x₂) ]	=	x₁ + 2 x₂
[s (x₁) ]	=	x₁ + 1
[U31# (x₁, x₂) ]	=	x₁ + 1
[and# (x₁, x₂) ]	=	x₁ + 1
[U21# (x₁, x₂) ]	=	x₁ + 2
[n__isNatKind (x₁) ]	=	x₁
[U11# (x₁, x₂, x₃) ]	=	x₁ + x₂ + 1
[n__s (x₁) ]	=	x₁ + 1
[plus# (x₁, x₂) ]	=	3 x₁ + 3 x₂ + 1
[U12# (x₁, x₂) ]	=	x₁ + 1
[tt]	=	0
[isNat (x₁) ]	=	x₁
[activate# (x₁) ]	=	x₁ + 1
[plus (x₁, x₂) ]	=	3 x₁ + 3 x₂
[n__and (x₁, x₂) ]	=	x₁ + 2 x₂
[n__isNat (x₁) ]	=	x₁
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n

[U11 (x₁, x₂, x₃) ]	=	2 x₁ + 3
[n__plus (x₁, x₂) ]	=	2 x₁ + 3 x₂ + 3
[isNatKind (x₁) ]	=	x₁
[U22 (x₁) ]	=	1
[n__0]	=	2
[U13 (x₁) ]	=	1
[U12 (x₁, x₂) ]	=	1
[0]	=	2
[U41 (x₁, x₂, x₃) ]	=	3 x₁ + 2 x₂ + 3
[isNat# (x₁) ]	=	2 x₁
[activate (x₁) ]	=	x₁
[U21 (x₁, x₂) ]	=	x₁ + 1
[isNatKind# (x₁) ]	=	2 x₁
[and (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[U31 (x₁, x₂) ]	=	x₁ + 3
[s (x₁) ]	=	x₁
[and# (x₁, x₂) ]	=	3 x₁ + 2
[U31# (x₁, x₂) ]	=	3 x₁ + 1
[n__isNatKind (x₁) ]	=	x₁
[U11# (x₁, x₂, x₃) ]	=	3 x₁ + 3 x₂
[n__s (x₁) ]	=	x₁
[plus# (x₁, x₂) ]	=	3 x₁ + 2 x₂
[U12# (x₁, x₂) ]	=	3 x₁
[tt]	=	0
[isNat (x₁) ]	=	2 x₁ + 1
[activate# (x₁) ]	=	3 x₁
[plus (x₁, x₂) ]	=	2 x₁ + 3 x₂ + 3
[n__and (x₁, x₂) ]	=	2 x₁ + x₂ + 1
[n__isNat (x₁) ]	=	2 x₁ + 1
[f(x₁, ..., x_n)]	=	x₁ + ... + x_n + 1	for all other symbols f of arity n