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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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:

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
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)))
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 01
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(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))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U121(tt, V2) → U131(isNat(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U211(tt, V1) → ACTIVATE(V1)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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:

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → AND(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))
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)))
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
ACTIVATE(n__0) → 01
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(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))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → S(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U111(tt, V1, V2) → ACTIVATE(V1)
U121(tt, V2) → U131(isNat(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U211(tt, V1) → ACTIVATE(V1)
U411(tt, M, N) → S(plus(activate(N), activate(M)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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 5 less nodes.

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

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

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, s(M)) → ISNAT(M)
U411(tt, M, N) → PLUS(activate(N), activate(M))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
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)))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, s(M)) → ISNAT(N)
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
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))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, M, N) → ACTIVATE(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U111(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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.


ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
PLUS(N, s(M)) → ISNAT(M)
U411(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)))
PLUS(N, s(M)) → ISNAT(N)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U411(tt, M, N) → ACTIVATE(N)
U411(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNAT(activate(V1))
PLUS(N, s(M)) → AND(isNat(M), n__isNatKind(M))
The remaining pairs can at least be oriented weakly.

U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U311(tt, N) → ACTIVATE(N)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U111(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x1, x2)) = (4)x_1 + (4)x_2   
POL(U121(x1, x2)) = x_2   
POL(U41(x1, x2, x3)) = 1 + (4)x_2 + (4)x_3   
POL(U211(x1, x2)) = 1 + x_2   
POL(activate(x1)) = x_1   
POL(U12(x1, x2)) = 0   
POL(and(x1, x2)) = (4)x_2   
POL(n__s(x1)) = 1 + x_1   
POL(U21(x1, x2)) = 0   
POL(PLUS(x1, x2)) = (4)x_1 + x_2   
POL(tt) = 0   
POL(U311(x1, x2)) = x_2   
POL(AND(x1, x2)) = x_2   
POL(s(x1)) = 1 + x_1   
POL(isNat(x1)) = 0   
POL(ACTIVATE(x1)) = x_1   
POL(plus(x1, x2)) = (4)x_1 + (4)x_2   
POL(U31(x1, x2)) = x_2   
POL(U111(x1, x2, x3)) = x_2 + x_3   
POL(ISNATKIND(x1)) = x_1   
POL(n__isNatKind(x1)) = x_1   
POL(U11(x1, x2, x3)) = 0   
POL(isNatKind(x1)) = x_1   
POL(0) = 0   
POL(U411(x1, x2, x3)) = 1 + x_2 + (4)x_3   
POL(ISNAT(x1)) = x_1   
POL(n__and(x1, x2)) = (4)x_2   
POL(U22(x1)) = 0   
POL(n__0) = 0   
POL(U13(x1)) = 0   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

activate(n__s(X)) → s(X)
activate(X) → X
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
s(X) → n__s(X)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
0 → n__0



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

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

U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U311(tt, N) → ACTIVATE(N)
U121(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U111(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, s(M)) → U411(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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 2 less nodes.

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

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

U111(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U111(tt, V1, V2) → ISNAT(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
U111(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U311(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U121(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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)
U121(tt, V2) → ACTIVATE(V2)
U111(tt, V1, V2) → ISNAT(activate(V1))
U111(tt, V1, V2) → ACTIVATE(V1)
U111(tt, V1, V2) → U121(isNat(activate(V1)), activate(V2))
U121(tt, V2) → ISNAT(activate(V2))
The remaining pairs can at least be oriented weakly.

ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U311(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x1, x2)) = (4)x_1 + (2)x_2   
POL(U121(x1, x2)) = 2 + (4)x_2   
POL(U41(x1, x2, x3)) = 4 + (2)x_2 + (4)x_3   
POL(activate(x1)) = x_1   
POL(U12(x1, x2)) = 4 + x_2   
POL(n__s(x1)) = 4 + x_1   
POL(and(x1, x2)) = x_1 + x_2   
POL(U21(x1, x2)) = 3 + x_2   
POL(PLUS(x1, x2)) = (4)x_1   
POL(tt) = 2   
POL(U311(x1, x2)) = (2)x_2   
POL(AND(x1, x2)) = (2)x_2   
POL(s(x1)) = 4 + x_1   
POL(isNat(x1)) = 2 + (3)x_1   
POL(ACTIVATE(x1)) = (2)x_1   
POL(plus(x1, x2)) = (4)x_1 + (2)x_2   
POL(U31(x1, x2)) = x_2   
POL(n__isNatKind(x1)) = x_1   
POL(ISNATKIND(x1)) = (2)x_1   
POL(U111(x1, x2, x3)) = (4)x_1 + (4)x_2 + (4)x_3   
POL(U11(x1, x2, x3)) = 1 + (3)x_1 + (3)x_3   
POL(isNatKind(x1)) = x_1   
POL(0) = 4   
POL(ISNAT(x1)) = (4)x_1   
POL(n__and(x1, x2)) = x_1 + x_2   
POL(U22(x1)) = 3   
POL(n__0) = 4   
POL(U13(x1)) = 2   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

activate(n__s(X)) → s(X)
activate(X) → X
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
s(X) → n__s(X)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
0 → n__0



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

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

ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U311(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → U111(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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 1 less node.

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

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

ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U311(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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.


U311(tt, N) → ACTIVATE(N)
The remaining pairs can at least be oriented weakly.

ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), N)
Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x1, x2)) = (3)x_1 + (2)x_2   
POL(U41(x1, x2, x3)) = (2)x_2 + (3)x_3   
POL(U12(x1, x2)) = 2 + x_2   
POL(activate(x1)) = x_1   
POL(n__s(x1)) = x_1   
POL(and(x1, x2)) = x_2   
POL(U21(x1, x2)) = (2)x_1 + (2)x_2   
POL(PLUS(x1, x2)) = (3)x_1   
POL(tt) = 2   
POL(U311(x1, x2)) = (2)x_1 + x_2   
POL(AND(x1, x2)) = x_2   
POL(s(x1)) = x_1   
POL(isNat(x1)) = (4)x_1   
POL(ACTIVATE(x1)) = x_1   
POL(plus(x1, x2)) = (3)x_1 + (2)x_2   
POL(U31(x1, x2)) = (2)x_1 + x_2   
POL(n__isNatKind(x1)) = x_1   
POL(ISNATKIND(x1)) = x_1   
POL(U11(x1, x2, x3)) = (4)x_1 + (2)x_2 + x_3   
POL(isNatKind(x1)) = x_1   
POL(0) = 2   
POL(ISNAT(x1)) = (2)x_1   
POL(n__and(x1, x2)) = x_2   
POL(U22(x1)) = 4   
POL(n__0) = 2   
POL(U13(x1)) = 2   
The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

activate(n__s(X)) → s(X)
activate(X) → X
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
s(X) → n__s(X)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
0 → n__0



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

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

ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
PLUS(N, 0) → U311(and(isNat(N), n__isNatKind(N)), 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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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 1 less node.

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

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

ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
AND(tt, X) → ACTIVATE(X)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))

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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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.


ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → AND(isNat(N), n__isNatKind(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
The remaining pairs can at least be oriented weakly.

ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
Used ordering: Polynomial interpretation [25,35]:

POL(n__plus(x1, x2)) = 1 + (4)x_1 + (2)x_2   
POL(U41(x1, x2, x3)) = 4 + (2)x_2 + (4)x_3   
POL(U12(x1, x2)) = (2)x_1   
POL(activate(x1)) = x_1   
POL(and(x1, x2)) = (2)x_1 + (2)x_2   
POL(n__s(x1)) = 2 + x_1   
POL(U21(x1, x2)) = 0   
POL(PLUS(x1, x2)) = 1 + (2)x_1   
POL(tt) = 0   
POL(AND(x1, x2)) = (2)x_1 + x_2   
POL(s(x1)) = 2 + x_1   
POL(isNat(x1)) = 0   
POL(ACTIVATE(x1)) = x_1   
POL(plus(x1, x2)) = 1 + (4)x_1 + (2)x_2   
POL(U31(x1, x2)) = 1 + (4)x_2   
POL(n__isNatKind(x1)) = (2)x_1   
POL(ISNATKIND(x1)) = (2)x_1   
POL(U11(x1, x2, x3)) = 0   
POL(isNatKind(x1)) = (2)x_1   
POL(0) = 0   
POL(ISNAT(x1)) = x_1   
POL(n__and(x1, x2)) = (2)x_1 + (2)x_2   
POL(U22(x1)) = 0   
POL(n__0) = 0   
POL(U13(x1)) = 0   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

activate(n__s(X)) → s(X)
activate(X) → X
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
isNatKind(n__s(V1)) → isNatKind(activate(V1))
and(tt, X) → activate(X)
U31(tt, N) → activate(N)
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
U13(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U11(tt, V1, V2) → U12(isNat(activate(V1)), activate(V2))
U12(tt, V2) → U13(isNat(activate(V2)))
U41(tt, M, N) → s(plus(activate(N), activate(M)))
U22(tt) → tt
isNatKind(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__plus(V1, V2)) → U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
plus(N, s(M)) → U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N)
s(X) → n__s(X)
isNatKind(X) → n__isNatKind(X)
plus(X1, X2) → n__plus(X1, X2)
0 → n__0



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

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

ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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 2 less nodes.

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

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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__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.


AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(n__and(x1, x2)) = 4 + (4)x_1 + (4)x_2   
POL(tt) = 4   
POL(AND(x1, x2)) = 1 + (4)x_1 + x_2   
POL(ACTIVATE(x1)) = 4 + x_1   
The value of delta used in the strict ordering is 7.
The following usable rules [17] were oriented: none



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

Q DP problem:
P is empty.
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, N) → activate(N)
U41(tt, M, N) → s(plus(activate(N), activate(M)))
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))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → and(isNatKind(activate(V1)), n__isNatKind(activate(V2)))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
plus(N, 0) → U31(and(isNat(N), n__isNatKind(N)), N)
plus(N, s(M)) → U41(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)
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__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.