Termination w.r.t. Q proof of /tmp/tmpRX7lF_/MYNAT_nokinds-noand

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, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(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:

U31¹(tt, V2) → U32¹(isNat(activate(V2)))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
X(N, 0) → ISNAT(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U52¹(tt, M, N) → S(plus(activate(N), activate(M)))
X(N, s(M)) → U71¹(isNat(M), M, N)
U72¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__0) → 0¹
X(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U51¹(tt, M, N) → ISNAT(activate(N))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
U52¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__s(X)) → S(activate(X))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U71¹(tt, M, N) → ACTIVATE(N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U31¹(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
U71¹(tt, M, N) → ISNAT(activate(N))
U11¹(tt, V2) → U12¹(isNat(activate(V2)))
U72¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
U31¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U11¹(tt, V2) → ACTIVATE(V2)
PLUS(N, 0) → ISNAT(N)
U61¹(tt) → 0¹
U72¹(tt, M, N) → X(activate(N), activate(M))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U52¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(isNat(N), N)
U11¹(tt, V2) → ISNAT(activate(V2))
X(N, 0) → U61¹(isNat(N))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(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:

U31¹(tt, V2) → U32¹(isNat(activate(V2)))
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
X(N, 0) → ISNAT(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U52¹(tt, M, N) → S(plus(activate(N), activate(M)))
X(N, s(M)) → U71¹(isNat(M), M, N)
U72¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__0) → 0¹
X(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U51¹(tt, M, N) → ISNAT(activate(N))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
U52¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__s(X)) → S(activate(X))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U71¹(tt, M, N) → ACTIVATE(N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U31¹(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
U71¹(tt, M, N) → ISNAT(activate(N))
U11¹(tt, V2) → U12¹(isNat(activate(V2)))
U72¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
U31¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U11¹(tt, V2) → ACTIVATE(V2)
PLUS(N, 0) → ISNAT(N)
U61¹(tt) → 0¹
U72¹(tt, M, N) → X(activate(N), activate(M))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U52¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → U41¹(isNat(N), N)
U11¹(tt, V2) → ISNAT(activate(V2))
X(N, 0) → U61¹(isNat(N))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(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:

U72¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U51¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
X(N, 0) → ISNAT(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U72¹(tt, M, N) → ACTIVATE(N)
U31¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
X(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U51¹(tt, M, N) → ISNAT(activate(N))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
U11¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
U52¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U52¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U71¹(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → U41¹(isNat(N), N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
U31¹(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
U71¹(tt, M, N) → ISNAT(activate(N))

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(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.

U72¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, M, N) → ACTIVATE(N)
U51¹(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → ISNAT(M)
ISNAT(n__x(V1, V2)) → U31¹(isNat(activate(V1)), activate(V2))
PLUS(N, s(M)) → U51¹(isNat(M), M, N)
X(N, 0) → ISNAT(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
X(N, s(M)) → U71¹(isNat(M), M, N)
U72¹(tt, M, N) → ACTIVATE(N)
U31¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
X(N, s(M)) → ISNAT(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U51¹(tt, M, N) → ISNAT(activate(N))
ISNAT(n__x(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
PLUS(N, 0) → ISNAT(N)
U11¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U71¹(tt, M, N) → ACTIVATE(M)
U72¹(tt, M, N) → X(activate(N), activate(M))
U52¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U52¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U71¹(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → U41¹(isNat(N), N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
U72¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
U31¹(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
U71¹(tt, M, N) → ISNAT(activate(N))

The remaining pairs can at least be oriented weakly.

U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
U41¹(tt, N) → ACTIVATE(N)
U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))

Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2, x3) = U72¹(x1, x2, x3)
tt = tt
ACTIVATE(x1) = ACTIVATE(x1)
U51¹(x1, x2, x3) = U51¹(x2, x3)
U52¹(x1, x2, x3) = U52¹(x2, x3)
isNat(x1) = isNat
activate(x1) = x1
PLUS(x1, x2) = PLUS(x1, x2)
s(x1) = s(x1)
ISNAT(x1) = x1
n__x(x1, x2) = n__x(x1, x2)
U31¹(x1, x2) = U31¹(x1, x2)
X(x1, x2) = X(x1, x2)
0 = 0
n__s(x1) = n__s(x1)
U71¹(x1, x2, x3) = U71¹(x1, x2, x3)
n__plus(x1, x2) = n__plus(x1, x2)
U11¹(x1, x2) = U11¹(x1, x2)
U41¹(x1, x2) = U41¹(x2)
x(x1, x2) = x(x1, x2)
U32(x1) = x1
U21(x1) = x1
U31(x1, x2) = x1
U11(x1, x2) = U11
U12(x1) = U12
n__0 = n__0
U61(x1) = U61
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2, x3) = U52(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U41(x1, x2) = U41(x1, x2)
U72(x1, x2, x3) = U72(x1, x2, x3)
U71(x1, x2, x3) = U71(x1, x2, x3)

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

[U72^1₃, n_{_x₂}, X₂, U71^1₃, x₂, U61, U72₃, U71₃] > U31^1₂ > [ACTIVATE₁, U41^1₁]
[U72^1₃, n_{_x₂}, X₂, U71^1₃, x₂, U61, U72₃, U71₃] > [n_{_plus₂}, U51₃, U52₃, plus₂] > [tt, isNat, U11, U12] > [s₁, n_{_s₁}] > [U51^1₂, U52^1₂, PLUS₂] > [ACTIVATE₁, U41^1₁]
[U72^1₃, n_{_x₂}, X₂, U71^1₃, x₂, U61, U72₃, U71₃] > [n_{_plus₂}, U51₃, U52₃, plus₂] > [tt, isNat, U11, U12] > [0, n_₀] > [ACTIVATE₁, U41^1₁]
[U72^1₃, n_{_x₂}, X₂, U71^1₃, x₂, U61, U72₃, U71₃] > [n_{_plus₂}, U51₃, U52₃, plus₂] > U11^1₂ > [ACTIVATE₁, U41^1₁]
[U72^1₃, n_{_x₂}, X₂, U71^1₃, x₂, U61, U72₃, U71₃] > [n_{_plus₂}, U51₃, U52₃, plus₂] > U41₂ > [ACTIVATE₁, U41^1₁]

Status:

n_{_plus₂}: [2,1]
U52^1₂: multiset
U51^1₂: multiset
U41^1₁: multiset
U11: multiset
x₂: [1,2]
n_{_s₁}: [1]
U71^1₃: [3,2,1]
isNat: multiset
PLUS₂: multiset
U61: []
tt: multiset
U11^1₂: multiset
U31^1₂: multiset
s₁: [1]
U51₃: [2,3,1]
ACTIVATE₁: multiset
plus₂: [2,1]
X₂: [1,2]
U12: multiset
U72^1₃: [3,2,1]
0: multiset
U52₃: [2,3,1]
U41₂: [1,2]
n_₀: multiset
n_{_x₂}: [1,2]
U72₃: [3,2,1]
U71₃: [3,2,1]

The following usable rules [17] were oriented:

U32(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__0) → tt
U61(tt) → 0
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
s(X) → n__s(X)
plus(X1, X2) → n__plus(X1, X2)
0 → n__0
x(N, 0) → U61(isNat(N))
plus(N, s(M)) → U51(isNat(M), M, N)
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
plus(N, 0) → U41(isNat(N), N)
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U41(tt, N) → activate(N)
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
x(N, s(M)) → U71(isNat(M), M, N)
activate(n__0) → 0
x(X1, X2) → n__x(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

U71¹(tt, M, N) → U72¹(isNat(activate(N)), activate(M), activate(N))
U51¹(tt, M, N) → U52¹(isNat(activate(N)), activate(M), activate(N))
U52¹(tt, M, N) → PLUS(activate(N), activate(M))
U41¹(tt, N) → ACTIVATE(N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(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 4 less nodes.