Termination w.r.t. Q proof of /tmp/tmphOaHYF/PEANO_complete

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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(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)
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__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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(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)
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__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U31¹(tt, N) → ACTIVATE(N)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 0¹
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U12¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U41¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(activate(X))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U12¹(tt, V2) → U13¹(isNat(activate(V2)))
U21¹(tt, V1) → ACTIVATE(V1)
U41¹(tt, M, N) → S(plus(activate(N), activate(M)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), 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(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)
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__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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U31¹(tt, N) → ACTIVATE(N)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 0¹
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U12¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U11¹(tt, V1, V2) → ISNAT(activate(V1))
U41¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(activate(X))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U12¹(tt, V2) → U13¹(isNat(activate(V2)))
U21¹(tt, V1) → ACTIVATE(V1)
U41¹(tt, M, N) → S(plus(activate(N), activate(M)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), 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(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)
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__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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → ISNAT(M)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
U31¹(tt, N) → ACTIVATE(N)
U11¹(tt, V1, V2) → U12¹(isNat(activate(V1)), activate(V2))
U12¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → U11¹(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ISNAT(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U41¹(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
PLUS(N, s(M)) → U41¹(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N)
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U31¹(and(isNat(N), n__isNatKind(N)), 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(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)
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__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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U12¹(tt, V2) → ACTIVATE(V2)
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
U31¹(tt, N) → ACTIVATE(N)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))

Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1
n__s(x1) = n__s(x1)
ISNATKIND(x1) = ISNATKIND(x1)
activate(x1) = x1
U11¹(x1, x2, x3) = U11¹(x1, x2, x3)
tt = tt
ACTIVATE(x1) = ACTIVATE(x1)
U12¹(x1, x2) = U12¹(x2)
PLUS(x1, x2) = PLUS(x1, x2)
s(x1) = s(x1)
U41¹(x1, x2, x3) = U41¹(x2, x3)
0 = 0
AND(x1, x2) = AND(x2)
isNat(x1) = x1
n__isNatKind(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
isNatKind(x1) = x1
n__and(x1, x2) = n__and(x1, x2)
U31¹(x1, x2) = U31¹(x2)
and(x1, x2) = and(x1, x2)
U21¹(x1, x2) = U21¹(x2)
n__isNat(x1) = x1
U12(x1, x2) = U12(x1, x2)
U13(x1) = U13(x1)
U11(x1, x2, x3) = U11(x2, x3)
U21(x1, x2) = U21(x2)
U22(x1) = x1
n__0 = n__0
U41(x1, x2, x3) = U41(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U31(x1, x2) = U31(x1, x2)

Recursive path order with status [2].
Quasi-Precedence:

[n_{_plus₂}, U41₃, plus₂] > [PLUS₂, U41^1₂] > [n_{_s₁}, ISNATKIND₁, U11^1₃, tt, ACTIVATE₁, U12^1₁, s₁, 0, AND₁, n_{_and₂}, U31^1₁, and₂, U21^1₁, U21₁, n_₀]
[n_{_plus₂}, U41₃, plus₂] > U11₂ > [U12₂, U13₁] > [n_{_s₁}, ISNATKIND₁, U11^1₃, tt, ACTIVATE₁, U12^1₁, s₁, 0, AND₁, n_{_and₂}, U31^1₁, and₂, U21^1₁, U21₁, n_₀]
[n_{_plus₂}, U41₃, plus₂] > U31₂ > [n_{_s₁}, ISNATKIND₁, U11^1₃, tt, ACTIVATE₁, U12^1₁, s₁, 0, AND₁, n_{_and₂}, U31^1₁, and₂, U21^1₁, U21₁, n_₀]

Status:

n_{_plus₂}: [2,1]
U41₃: [2,3,1]
U31^1₁: multiset
U12₂: [2,1]
n_{_s₁}: multiset
and₂: multiset
PLUS₂: multiset
tt: multiset
s₁: multiset
ACTIVATE₁: multiset
plus₂: [2,1]
U31₂: multiset
AND₁: multiset
ISNATKIND₁: multiset
U11^1₃: multiset
U11₂: multiset
0: multiset
n_{_and₂}: multiset
U12^1₁: multiset
U41^1₂: multiset
n_₀: multiset
U13₁: [1]
U21^1₁: multiset
U21₁: multiset

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

U31¹(tt, N) → ACTIVATE(N)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U12¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U41¹(tt, M, N) → PLUS(activate(N), activate(M))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U21¹(tt, V1) → 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(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)
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__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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 9 less nodes.