Termination w.r.t. Q proof of /tmp/tmpDIcF8I/MYNAT_nokinds

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(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:

X(N, 0) → U31¹(isNat(N))
U41¹(tt, M, N) → X(activate(N), activate(M))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
U21¹(tt, M, N) → S(plus(activate(N), activate(M)))
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
PLUS(N, 0) → U11¹(isNat(N), N)
X(N, 0) → ISNAT(N)
X(N, s(M)) → U41¹(and(isNat(M), n__isNat(N)), M, N)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(N, s(M)) → ISNAT(M)
U41¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
U11¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
U41¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U41¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__s(X)) → S(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U31¹(tt) → 0¹
U21¹(tt, M, N) → ACTIVATE(M)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
X(N, s(M)) → AND(isNat(M), n__isNat(N))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(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:

X(N, 0) → U31¹(isNat(N))
U41¹(tt, M, N) → X(activate(N), activate(M))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
U21¹(tt, M, N) → S(plus(activate(N), activate(M)))
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
PLUS(N, 0) → U11¹(isNat(N), N)
X(N, 0) → ISNAT(N)
X(N, s(M)) → U41¹(and(isNat(M), n__isNat(N)), M, N)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(N, s(M)) → ISNAT(M)
U41¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
U11¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
U41¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U41¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__s(X)) → S(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U31¹(tt) → 0¹
U21¹(tt, M, N) → ACTIVATE(M)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
X(N, s(M)) → AND(isNat(M), n__isNat(N))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

U41¹(tt, M, N) → X(activate(N), activate(M))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))
PLUS(N, 0) → U11¹(isNat(N), N)
X(N, 0) → ISNAT(N)
X(N, s(M)) → U41¹(and(isNat(M), n__isNat(N)), M, N)
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
U41¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNat(N))
U11¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
U41¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U41¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U21¹(tt, M, N) → ACTIVATE(M)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U21¹(and(isNat(M), n__isNat(N)), M, N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U21¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__isNat(X)) → ISNAT(X)
X(N, s(M)) → AND(isNat(M), n__isNat(N))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__x(V1, V2)) → AND(isNat(activate(V1)), n__isNat(activate(V2)))

The TRS R consists of the following rules:

U11(tt, N) → activate(N)
U21(tt, M, N) → s(plus(activate(N), activate(M)))
U31(tt) → 0
U41(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)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
isNat(n__s(V1)) → isNat(activate(V1))
isNat(n__x(V1, V2)) → and(isNat(activate(V1)), n__isNat(activate(V2)))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), n__isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), n__isNat(N)), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
isNat(X) → n__isNat(X)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNat(X)) → isNat(X)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

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