(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → U131(isNat(activate(V2)))
U121(tt, V2) → ISNAT(activate(V2))
U121(tt, V2) → ACTIVATE(V2)
U211(tt, V1) → U221(isNat(activate(V1)))
U211(tt, V1) → ISNAT(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U311(tt, N) → ACTIVATE(N)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
U411(tt, M, N) → PLUS(activate(N), activate(M))
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → U111(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__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(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) → U311(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)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
ACTIVATE(n__0) → 01
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
U311(tt, N) → ACTIVATE(N)
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__and(X1, X2)) → AND(X1, X2)
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) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
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__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
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)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
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)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
PLUS(N, s(M)) → ISNAT(N)
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x1, x2)
tt  =  tt
ISNAT(x1)  =  ISNAT(x1)
activate(x1)  =  x1
n__plus(x1, x2)  =  n__plus(x1, x2)
U111(x1, x2, x3)  =  U111(x1, x2, x3)
and(x1, x2)  =  x2
isNatKind(x1)  =  x1
n__isNatKind(x1)  =  x1
isNat(x1)  =  isNat
ACTIVATE(x1)  =  ACTIVATE(x1)
PLUS(x1, x2)  =  PLUS(x1, x2)
0  =  0
U311(x1, x2)  =  U311(x2)
ISNATKIND(x1)  =  ISNATKIND(x1)
AND(x1, x2)  =  AND(x2)
n__and(x1, x2)  =  x2
n__s(x1)  =  x1
U211(x1, x2)  =  U211(x2)
s(x1)  =  x1
U411(x1, x2, x3)  =  U411(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
U41(x1, x2, x3)  =  U41(x2, x3)
n__0  =  n__0
U31(x1, x2)  =  U31(x2)
U11(x1, x2, x3)  =  U11
U21(x1, x2)  =  U21
U12(x1, x2)  =  U12
U13(x1)  =  x1
U22(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
[0, n0] > [tt, isNat, U11, U21, U12] > [U12^12, ISNAT1, U11^13, ACTIVATE1, PLUS2, U31^11, ISNATKIND1, AND1, U21^11, U41^12] > U311
[0, n0] > [tt, isNat, U11, U21, U12] > [nplus2, plus2, U412] > U311


The following usable rules [FROCOS05] were oriented:

activate(X) → X
activate(n__s(X)) → s(X)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U311(tt, N) → ACTIVATE(N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U411(tt, M, N) → PLUS(activate(N), activate(M))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(tt, M, N) → PLUS(activate(N), activate(M))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2, x3)  =  U411(x2)
tt  =  tt
PLUS(x1, x2)  =  x2
activate(x1)  =  x1
s(x1)  =  s(x1)
and(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
n__isNatKind(x1)  =  n__isNatKind
n__and(x1, x2)  =  x2
plus(x1, x2)  =  plus(x1, x2)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
0  =  0
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
isNatKind(x1)  =  isNatKind
n__s(x1)  =  n__s(x1)
U31(x1, x2)  =  U31(x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U21(x1, x2)  =  U21(x1, x2)
U12(x1, x2)  =  U12
U13(x1)  =  U13
U22(x1)  =  U22

Recursive Path Order [RPO].
Precedence:
isNat1 > [plus2, U413, nplus2, U113, U12, U13] > [U41^11, tt, s1, nisNatKind, isNatKind, ns1]
isNat1 > [plus2, U413, nplus2, U113, U12, U13] > U311
isNat1 > U212 > U22 > [U41^11, tt, s1, nisNatKind, isNatKind, ns1]
[0, n0] > [U41^11, tt, s1, nisNatKind, isNatKind, ns1]


The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(13) TRUE

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(n__s(V1)) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
n__s(x1)  =  x1
activate(x1)  =  x1
ACTIVATE(x1)  =  x1
n__isNatKind(x1)  =  n__isNatKind(x1)
n__and(x1, x2)  =  x2
AND(x1, x2)  =  x2
tt  =  tt
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  x1
U41(x1, x2, x3)  =  U41(x2, x3)
and(x1, x2)  =  x2
isNat(x1)  =  x1
0  =  0
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
isNatKind(x1)  =  isNatKind(x1)
U31(x1, x2)  =  U31(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U21(x1, x2)  =  x2
U12(x1, x2)  =  U12(x1, x2)
U13(x1)  =  x1
U22(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
[plus2, U412, nplus2] > [ISNATKIND1, nisNatKind1, isNatKind1]
[plus2, U412, nplus2] > U312
[plus2, U412, nplus2] > U113 > U122
[0, n0] > [ISNATKIND1, nisNatKind1, isNatKind1]
[0, n0] > tt > U122
[0, n0] > U312


The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(18) Complex Obligation (AND)

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
n__s(x1)  =  n__s(x1)
activate(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
and(x1, x2)  =  x2
isNat(x1)  =  isNat
n__isNatKind(x1)  =  n__isNatKind
n__and(x1, x2)  =  x2
0  =  0
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
isNatKind(x1)  =  isNatKind
tt  =  tt
U31(x1, x2)  =  x2
U11(x1, x2, x3)  =  U11
U21(x1, x2)  =  x1
U12(x1, x2)  =  x1
U13(x1)  =  U13
U22(x1)  =  U22

Recursive Path Order [RPO].
Precedence:
[plus2, U413, nplus2] > [isNat, U11] > [ns1, s1, nisNatKind, isNatKind] > [tt, U13, U22]
[0, n0] > [isNat, U11] > [ns1, s1, nisNatKind, isNatKind] > [tt, U13, U22]


The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(21) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(23) TRUE

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1)  =  ACTIVATE(x1)
n__and(x1, x2)  =  n__and(x2)
AND(x1, x2)  =  AND(x2)
tt  =  tt

Recursive Path Order [RPO].
Precedence:
nand1 > [ACTIVATE1, AND1, tt]


The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(tt, X) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(28) TRUE

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
n__s(x1)  =  n__s(x1)
U211(x1, x2)  =  U211(x1, x2)
isNatKind(x1)  =  isNatKind
activate(x1)  =  x1
tt  =  tt
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
and(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
n__isNatKind(x1)  =  n__isNatKind
n__and(x1, x2)  =  x2
0  =  0
n__0  =  n__0
n__plus(x1, x2)  =  n__plus(x1, x2)
U31(x1, x2)  =  x2
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U21(x1, x2)  =  x2
U12(x1, x2)  =  U12(x2)
U13(x1)  =  U13
U22(x1)  =  U22

Recursive Path Order [RPO].
Precedence:
[ISNAT1, U21^12] > [isNatKind, nisNatKind] > [tt, 0, n0, U22]
[plus2, U413, nplus2] > [ns1, s1] > [isNatKind, nisNatKind] > [tt, 0, n0, U22]
isNat1 > [isNatKind, nisNatKind] > [tt, 0, n0, U22]
isNat1 > U113 > U121 > U13 > [tt, 0, n0, U22]


The following usable rules [FROCOS05] were oriented:

plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U13(tt) → tt
U12(tt, V2) → U13(isNat(activate(V2)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
activate(n__s(X)) → s(X)

(31) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(32) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(33) TRUE