Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 QDPOrderProof (⇔)
↳6 QDP
↳7 PisEmptyProof (⇔)
↳8 TRUE

(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, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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, V1, V2) → U321(isNat(activate(V1)), activate(V2))
U311(tt, V1, V2) → ISNAT(activate(V1))
U311(tt, V1, V2) → ACTIVATE(V1)
U311(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V2) → U331(isNat(activate(V2)))
U321(tt, V2) → ISNAT(activate(V2))
U321(tt, V2) → ACTIVATE(V2)
U411(tt, N) → ACTIVATE(N)
U511(tt, M, N) → S(plus(activate(N), activate(M)))
U511(tt, M, N) → PLUS(activate(N), activate(M))
U511(tt, M, N) → ACTIVATE(N)
U511(tt, M, N) → ACTIVATE(M)
U611(tt) → 01
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U711(tt, M, N) → X(activate(N), activate(M))
U711(tt, M, N) → ACTIVATE(N)
U711(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)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
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)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → U411(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)) → U511(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)
X(N, 0) → U611(and(isNat(N), n__isNatKind(N)))
X(N, 0) → AND(isNat(N), n__isNatKind(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 01
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__x(X1, X2)) → X(activate(X1), activate(X2))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(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, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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 8 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(activate(X1), activate(X2))
PLUS(N, 0) → U411(and(isNat(N), n__isNatKind(N)), N)
U411(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__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → AND(isNat(N), n__isNatKind(N))
X(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))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
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)) → 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))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U311(tt, V1, V2) → U321(isNat(activate(V1)), activate(V2))
U321(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U321(tt, V2) → ACTIVATE(V2)
U311(tt, V1, V2) → ISNAT(activate(V1))
U311(tt, V1, V2) → ACTIVATE(V1)
U311(tt, V1, V2) → ACTIVATE(V2)
U211(tt, V1) → ACTIVATE(V1)
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U511(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U511(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)
U511(tt, M, N) → ACTIVATE(N)
U511(tt, M, N) → ACTIVATE(M)
U711(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
X(N, s(M)) → ISNAT(M)
U711(tt, M, N) → ACTIVATE(N)
U711(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, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
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(activate(X1), activate(X2))
PLUS(N, 0) → U411(and(isNat(N), n__isNatKind(N)), N)
U411(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__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → AND(isNat(N), n__isNatKind(N))
X(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))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
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)) → 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))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U311(tt, V1, V2) → U321(isNat(activate(V1)), activate(V2))
U321(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U321(tt, V2) → ACTIVATE(V2)
U311(tt, V1, V2) → ISNAT(activate(V1))
U311(tt, V1, V2) → ACTIVATE(V1)
U311(tt, V1, V2) → ACTIVATE(V2)
U211(tt, V1) → ACTIVATE(V1)
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U511(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U511(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)
U511(tt, M, N) → ACTIVATE(N)
U511(tt, M, N) → ACTIVATE(M)
U711(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
X(N, s(M)) → ISNAT(M)
U711(tt, M, N) → ACTIVATE(N)
U711(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
U121(x0, x1, x2)  =  U121(x0)
ISNAT(x0, x1)  =  ISNAT(x1)
U111(x0, x1, x2, x3)  =  U111(x0)
ACTIVATE(x0, x1)  =  ACTIVATE(x0, x1)
PLUS(x0, x1, x2)  =  PLUS(x0)
U411(x0, x1, x2)  =  U411(x2)
ISNATKIND(x0, x1)  =  ISNATKIND(x1)
AND(x0, x1, x2)  =  AND(x0)
X(x0, x1, x2)  =  X(x0)
U211(x0, x1, x2)  =  U211(x0, x2)
U311(x0, x1, x2, x3)  =  U311(x0)
U321(x0, x1, x2)  =  U321(x0, x2)
U711(x0, x1, x2, x3)  =  U711(x0)
U511(x0, x1, x2, x3)  =  U511(x0)

Tags:
U121 has argument tags [12,48,0] and root tag 1
ISNAT has argument tags [52,16] and root tag 2
U111 has argument tags [44,15,14,46] and root tag 10
ACTIVATE has argument tags [33,16] and root tag 4
PLUS has argument tags [0,11,31] and root tag 0
U411 has argument tags [16,30,16] and root tag 8
ISNATKIND has argument tags [46,8] and root tag 10
AND has argument tags [21,31,15] and root tag 8
X has argument tags [0,0,50] and root tag 6
U211 has argument tags [16,25,24] and root tag 8
U311 has argument tags [4,10,0,12] and root tag 0
U321 has argument tags [16,45,16] and root tag 5
U711 has argument tags [28,0,8,0] and root tag 8
U511 has argument tags [0,63,32,47] and root tag 8

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
U121(x1, x2)  =  U121(x1, x2)
tt  =  tt
ISNAT(x1)  =  ISNAT
activate(x1)  =  x1
n__plus(x1, x2)  =  n__plus(x1, x2)
U111(x1, x2, x3)  =  U111(x2, x3)
and(x1, x2)  =  and(x1, x2)
isNatKind(x1)  =  x1
n__isNatKind(x1)  =  x1
isNat(x1)  =  x1
ACTIVATE(x1)  =  ACTIVATE(x1)
PLUS(x1, x2)  =  PLUS(x1, x2)
0  =  0
U411(x1, x2)  =  U411(x1)
ISNATKIND(x1)  =  ISNATKIND
AND(x1, x2)  =  x2
n__s(x1)  =  n__s(x1)
n__x(x1, x2)  =  n__x(x1, x2)
X(x1, x2)  =  X(x1, x2)
n__and(x1, x2)  =  n__and(x1, x2)
n__isNat(x1)  =  x1
U211(x1, x2)  =  x2
U311(x1, x2, x3)  =  U311(x2, x3)
U321(x1, x2)  =  x2
s(x1)  =  s(x1)
U711(x1, x2, x3)  =  U711(x1, x2, x3)
x(x1, x2)  =  x(x1, x2)
U511(x1, x2, x3)  =  U511(x2, x3)
n__0  =  n__0
plus(x1, x2)  =  plus(x1, x2)
U41(x1, x2)  =  U41(x1, x2)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U11(x1, x2, x3)  =  U11(x2, x3)
U21(x1, x2)  =  x2
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U61(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U12(x1, x2)  =  U12(x2)
U22(x1)  =  x1
U32(x1, x2)  =  U32
U13(x1)  =  U13(x1)
U33(x1)  =  U33

Lexicographic path order with status [LPO].
Quasi-Precedence:
U41^11 > ACTIVATE1 > ISNAT > [and2, ns1, nand2, s1]
U41^11 > ACTIVATE1 > ISNAT > ISNATKIND
[nx2, X2, U71^13, x2, U713] > [nplus2, PLUS2, U51^12, plus2, U112, U513] > [U12^12, U11^12] > ISNAT > [and2, ns1, nand2, s1]
[nx2, X2, U71^13, x2, U713] > [nplus2, PLUS2, U51^12, plus2, U112, U513] > [U12^12, U11^12] > ISNAT > ISNATKIND
[nx2, X2, U71^13, x2, U713] > [nplus2, PLUS2, U51^12, plus2, U112, U513] > ACTIVATE1 > ISNAT > [and2, ns1, nand2, s1]
[nx2, X2, U71^13, x2, U713] > [nplus2, PLUS2, U51^12, plus2, U112, U513] > ACTIVATE1 > ISNAT > ISNATKIND
[nx2, X2, U71^13, x2, U713] > [nplus2, PLUS2, U51^12, plus2, U112, U513] > U412
[nx2, X2, U71^13, x2, U713] > [nplus2, PLUS2, U51^12, plus2, U112, U513] > [U121, U131]
[nx2, X2, U71^13, x2, U713] > U31^12 > ISNAT > [and2, ns1, nand2, s1]
[nx2, X2, U71^13, x2, U713] > U31^12 > ISNAT > ISNATKIND
[nx2, X2, U71^13, x2, U713] > U313 > [U32, U33] > [tt, 0, n0] > [and2, ns1, nand2, s1]
[nx2, X2, U71^13, x2, U713] > U313 > [U32, U33] > [tt, 0, n0] > [U121, U131]

Status:
U12^12: [1,2]
tt: []
ISNAT: []
nplus2: [2,1]
U11^12: [1,2]
and2: [1,2]
ACTIVATE1: [1]
PLUS2: [2,1]
0: []
U41^11: [1]
ISNATKIND: []
ns1: [1]
nx2: [1,2]
X2: [1,2]
nand2: [1,2]
U31^12: [1,2]
s1: [1]
U71^13: [3,2,1]
x2: [1,2]
U51^12: [1,2]
n0: []
plus2: [2,1]
U412: [1,2]
U713: [3,2,1]
U112: [2,1]
U313: [3,2,1]
U513: [2,3,1]
U121: [1]
U32: []
U131: [1]
U33: []


The following usable rules [FROCOS05] were oriented:

activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
U41(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__x(X1, X2)) → x(activate(X1), activate(X2))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__isNat(X)) → isNat(X)
activate(X) → X
isNatKind(n__0) → tt
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(X) → n__isNat(X)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(X1, X2) → n__x(X1, X2)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U21(tt, V1) → U22(isNat(activate(V1)))
U31(tt, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U12(tt, V2) → U13(isNat(activate(V2)))
U32(tt, V2) → U33(isNat(activate(V2)))
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
U22(tt) → tt
U33(tt) → tt
U13(tt) → tt
U61(tt) → 0
0 → n__0

(6) 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, V1, V2) → U32(isNat(activate(V1)), activate(V2))
U32(tt, V2) → U33(isNat(activate(V2)))
U33(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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))
isNat(n__x(V1, V2)) → U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__x(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
plus(N, 0) → U41(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), n__isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNatKind(X) → n__isNatKind(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE