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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
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:

U111(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U511(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
U511(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)))
X(N, 0) → ISNAT(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 01
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))
X(N, s(M)) → ISNAT(M)
U311(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
U411(tt, N) → ACTIVATE(N)
U711(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U711(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, s(M)) → ISNAT(M)
U511(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, s(M)) → ISNAT(N)
U211(tt, V1) → U221(isNat(activate(V1)))
X(N, s(M)) → ISNAT(N)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
U511(tt, M, N) → S(plus(activate(N), activate(M)))
U311(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
U611(tt) → 01
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
U111(tt, V1, V2) → ISNAT(activate(V1))
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U321(tt, V2) → ISNAT(activate(V2))
U311(tt, V1, V2) → U321(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U121(tt, V2) → U131(isNat(activate(V2)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U211(tt, V1) → ACTIVATE(V1)
X(N, 0) → U611(and(isNat(N), n__isNatKind(N)))
U311(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U411(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U711(tt, M, N) → X(activate(N), activate(M))
U321(tt, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
U321(tt, V2) → U331(isNat(activate(V2)))
PLUS(N, s(M)) → U511(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
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:

U111(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U511(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
U511(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)))
X(N, 0) → ISNAT(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 01
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))
X(N, s(M)) → ISNAT(M)
U311(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
U411(tt, N) → ACTIVATE(N)
U711(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U711(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, s(M)) → ISNAT(M)
U511(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, s(M)) → ISNAT(N)
U211(tt, V1) → U221(isNat(activate(V1)))
X(N, s(M)) → ISNAT(N)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
U511(tt, M, N) → S(plus(activate(N), activate(M)))
U311(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
U611(tt) → 01
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
U111(tt, V1, V2) → ISNAT(activate(V1))
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U321(tt, V2) → ISNAT(activate(V2))
U311(tt, V1, V2) → U321(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U121(tt, V2) → U131(isNat(activate(V2)))
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U211(tt, V1) → ACTIVATE(V1)
X(N, 0) → U611(and(isNat(N), n__isNatKind(N)))
U311(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U411(and(isNat(N), n__isNatKind(N)), N)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U711(tt, M, N) → X(activate(N), activate(M))
U321(tt, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
U321(tt, V2) → U331(isNat(activate(V2)))
PLUS(N, s(M)) → U511(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
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 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof

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

U111(tt, V1, V2) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U511(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
U511(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)))
X(N, 0) → ISNAT(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), 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))
X(N, s(M)) → ISNAT(M)
U311(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
U411(tt, N) → ACTIVATE(N)
U711(tt, M, N) → ACTIVATE(M)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U711(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
U511(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, s(M)) → ISNAT(N)
X(N, s(M)) → ISNAT(N)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
U311(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U311(tt, V1, V2) → U321(isNat(activate(V1)), activate(V2))
U321(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U211(tt, V1) → ACTIVATE(V1)
U311(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U411(and(isNat(N), n__isNatKind(N)), N)
U711(tt, M, N) → X(activate(N), activate(M))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U321(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
PLUS(N, s(M)) → U511(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
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)
ISNAT(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U511(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
X(N, 0) → ISNAT(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
X(N, s(M)) → ISNAT(M)
U311(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
U711(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U711(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
U511(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, s(M)) → ISNAT(N)
X(N, s(M)) → ISNAT(N)
ISNAT(n__x(V1, V2)) → U311(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
U311(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → U711(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
X(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
X(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
U711(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U311(tt, V1, V2) → U321(isNat(activate(V1)), activate(V2))
U321(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
X(N, s(M)) → AND(isNat(M), n__isNatKind(M))
U311(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__x(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U411(and(isNat(N), n__isNatKind(N)), N)
U711(tt, M, N) → X(activate(N), activate(M))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U321(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
PLUS(N, s(M)) → U511(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
The remaining pairs can at least be oriented weakly.

U511(tt, M, N) → PLUS(activate(N), activate(M))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
U411(tt, N) → ACTIVATE(N)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U211(tt, V1) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
U211(tt, V1) → ACTIVATE(V1)
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  U111(x2, x3)
tt  =  tt
ACTIVATE(x1)  =  x1
ISNAT(x1)  =  x1
n__x(x1, x2)  =  n__x(x1, x2)
AND(x1, x2)  =  x2
isNatKind(x1)  =  isNatKind(x1)
activate(x1)  =  x1
n__isNatKind(x1)  =  n__isNatKind(x1)
U511(x1, x2, x3)  =  U511(x2, x3)
PLUS(x1, x2)  =  PLUS(x1, x2)
0  =  0
isNat(x1)  =  isNat
s(x1)  =  s(x1)
and(x1, x2)  =  x2
n__and(x1, x2)  =  x2
X(x1, x2)  =  X(x1, x2)
U121(x1, x2)  =  U121(x2)
ISNATKIND(x1)  =  x1
n__s(x1)  =  n__s(x1)
n__plus(x1, x2)  =  n__plus(x1, x2)
U311(x1, x2, x3)  =  U311(x1, x2, x3)
U411(x1, x2)  =  x2
U711(x1, x2, x3)  =  U711(x1, x2, x3)
U211(x1, x2)  =  x2
x(x1, x2)  =  x(x1, x2)
U321(x1, x2)  =  U321(x1, x2)
U32(x1, x2)  =  x1
U33(x1)  =  x1
U31(x1, x2, x3)  =  U31
U61(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
U11(x1, x2, x3)  =  U11
U12(x1, x2)  =  x1
U13(x1)  =  x1
U22(x1)  =  U22
U21(x1, x2)  =  U21
n__0  =  n__0
U71(x1, x2, x3)  =  U71(x1, x2, x3)
U41(x1, x2)  =  U41(x1, x2)

Recursive path order with status [2].
Quasi-Precedence:
[tt, 0, isNat, U31, U11, U22, U21, n0] > [nx2, X2, U71^13, x2, U713] > [U11^12, U51^12, PLUS2, nplus2, U513, plus2] > [isNatKind1, nisNatKind1] > [s1, ns1, U412]
[tt, 0, isNat, U31, U11, U22, U21, n0] > [nx2, X2, U71^13, x2, U713] > [U11^12, U51^12, PLUS2, nplus2, U513, plus2] > U12^11 > [s1, ns1, U412]
[tt, 0, isNat, U31, U11, U22, U21, n0] > [nx2, X2, U71^13, x2, U713] > U31^13 > U32^12 > [s1, ns1, U412]

Status:
nplus2: [2,1]
U51^12: [1,2]
U22: multiset
U31: multiset
U11: multiset
x2: [2,1]
ns1: multiset
U71^13: [2,3,1]
isNat: multiset
PLUS2: [2,1]
tt: multiset
U11^12: [2,1]
s1: multiset
U513: [2,3,1]
U21: multiset
plus2: [2,1]
X2: [2,1]
U31^13: multiset
nisNatKind1: multiset
U32^12: [1,2]
isNatKind1: multiset
0: multiset
U12^11: multiset
U412: multiset
n0: multiset
nx2: [2,1]
U713: [2,3,1]


The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof

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

ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
U211(tt, V1) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
U511(tt, M, N) → PLUS(activate(N), activate(M))
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U411(tt, N) → ACTIVATE(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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
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 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ UsableRulesProof

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

AND(tt, X) → ACTIVATE(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, 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(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(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)
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__x(X1, X2)) → x(X1, X2)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ UsableRulesProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: