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(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
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, 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(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
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:

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

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


U411(tt, M, N) → X(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)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
X(N, s(M)) → U411(and(isNat(M), n__isNat(N)), M, N)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(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))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
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))
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)
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)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
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.

ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
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
PLUS(x1, x2)  =  PLUS(x1, x2)
s(x1)  =  s(x1)
ISNAT(x1)  =  ISNAT(x1)
n__plus(x1, x2)  =  n__plus(x1, x2)
AND(x1, x2)  =  AND(x2)
isNat(x1)  =  isNat(x1)
n__isNat(x1)  =  n__isNat(x1)
0  =  0
U111(x1, x2)  =  U111(x1, x2)
ACTIVATE(x1)  =  x1
n__s(x1)  =  n__s(x1)
n__x(x1, x2)  =  n__x(x1, x2)
and(x1, x2)  =  x2
x(x1, x2)  =  x(x1, x2)
U211(x1, x2, x3)  =  U211(x1, x2, x3)
n__0  =  n__0
U31(x1)  =  U31(x1)
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(x2)

Recursive path order with status [2].
Quasi-Precedence:
[U41^13, X2, nx2, x2, U413] > [nplus2, U213, plus2, U111] > [PLUS2, isNat1, nisNat1, U21^13] > [tt, s1, ISNAT1, ns1] > AND1
[U41^13, X2, nx2, x2, U413] > [nplus2, U213, plus2, U111] > [PLUS2, isNat1, nisNat1, U21^13] > [tt, s1, ISNAT1, ns1] > [0, n0] > U11^12
[U41^13, X2, nx2, x2, U413] > U311 > [0, n0] > U11^12

Status:
nplus2: [1,2]
X2: [2,1]
plus2: [1,2]
AND1: multiset
nisNat1: [1]
U413: [2,3,1]
x2: [2,1]
ns1: multiset
U41^13: [2,3,1]
0: multiset
ISNAT1: multiset
PLUS2: [1,2]
tt: multiset
U11^12: [1,2]
U311: [1]
n0: multiset
nx2: [2,1]
s1: multiset
U111: [1]
U21^13: [3,2,1]
isNat1: [1]
U213: [3,2,1]


The following usable rules [17] were oriented:

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



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

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

ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))

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(activate(X1), activate(X2))
activate(n__isNat(X)) → isNat(X)
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 1 less node.