Termination w.r.t. Q proof of /tmp/tmpEK3K7v/MYNAT_nokinds

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, 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

          ↳ QDPOrderProof

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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))
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)
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)
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 remaining pairs can at least be oriented weakly.

U11¹(tt, N) → ACTIVATE(N)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__isNat(X)) → ISNAT(X)

Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3) = U41¹(x1, x2, x3)
tt = tt
X(x1, x2) = X(x1, x2)
activate(x1) = x1
ACTIVATE(x1) = ACTIVATE(x1)
n__plus(x1, x2) = n__plus(x1, x2)
PLUS(x1, x2) = PLUS(x1, x2)
s(x1) = s(x1)
ISNAT(x1) = ISNAT(x1)
AND(x1, x2) = AND(x1, x2)
isNat(x1) = x1
n__isNat(x1) = x1
0 = 0
U11¹(x1, x2) = U11¹(x2)
and(x1, x2) = and(x1, x2)
n__x(x1, x2) = n__x(x1, x2)
x(x1, x2) = x(x1, x2)
n__s(x1) = n__s(x1)
U21¹(x1, x2, x3) = U21¹(x2, x3)
U31(x1) = U31
n__0 = n__0
U21(x1, x2, x3) = U21(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U41(x1, x2, x3) = U41(x1, x2, x3)
U11(x1, x2) = U11(x1, x2)

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

[U41^1₃, X₂, n_{_x₂}, x₂, U41₃] > [n_{_plus₂}, U21₃, plus₂] > [ACTIVATE₁, PLUS₂, ISNAT₁, AND₂, U11^1₁, U21^1₂] > U11₂
[U41^1₃, X₂, n_{_x₂}, x₂, U41₃] > [n_{_plus₂}, U21₃, plus₂] > [s₁, n_{_s₁}] > U11₂
[U41^1₃, X₂, n_{_x₂}, x₂, U41₃] > [n_{_plus₂}, U21₃, plus₂] > and₂ > U11₂
[0, U31, n_₀] > tt > [ACTIVATE₁, PLUS₂, ISNAT₁, AND₂, U11^1₁, U21^1₂] > U11₂
[0, U31, n_₀] > tt > [s₁, n_{_s₁}] > U11₂

Status:

n_{_plus₂}: [1,2]
X₂: [1,2]
plus₂: [1,2]
U11^1₁: multiset
U31: multiset
U41₃: [3,2,1]
U21^1₂: multiset
x₂: [1,2]
and₂: multiset
n_{_s₁}: [1]
U11₂: [1,2]
U41^1₃: [3,2,1]
0: multiset
ISNAT₁: multiset
PLUS₂: multiset
tt: multiset
AND₂: multiset
n_₀: multiset
n_{_x₂}: [1,2]
s₁: [1]
U21₃: [3,2,1]
ACTIVATE₁: multiset

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

ACTIVATE(n__isNat(X)) → ISNAT(X)
U11¹(tt, N) → ACTIVATE(N)
U21¹(tt, M, N) → PLUS(activate(N), activate(M))

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 0 SCCs with 3 less nodes.