Termination w.r.t. Q proof of /tmp/tmpiAZ9om/PEANO_complete-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, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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:

U11¹(tt, V1, V2) → ACTIVATE(V2)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → U41¹(isNatKind(activate(V1)))
U21¹(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U62¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__0) → 0¹
U15¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U12¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U64¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U14¹(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → S(activate(X))
U61¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V2) → U32¹(isNatKind(activate(V2)))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U61¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U52¹(tt, N) → ACTIVATE(N)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U31¹(tt, V2) → ISNATKIND(activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, M, N) → ISNATKIND(activate(M))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U22¹(tt, V1) → U23¹(isNat(activate(V1)))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U64¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → S(plus(activate(N), activate(M)))
U63¹(tt, M, N) → ACTIVATE(N)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
U51¹(tt, N) → ACTIVATE(N)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U12¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
PLUS(N, 0) → ISNAT(N)
U14¹(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, M, N) → ISNAT(activate(N))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U22¹(tt, V1) → ACTIVATE(V1)
U13¹(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U62¹(tt, M, N) → ACTIVATE(N)
U15¹(tt, V2) → U16¹(isNat(activate(V2)))
U21¹(tt, V1) → ACTIVATE(V1)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U63¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U13¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U51¹(isNat(N), N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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:

U11¹(tt, V1, V2) → ACTIVATE(V2)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → U41¹(isNatKind(activate(V1)))
U21¹(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U62¹(tt, M, N) → ACTIVATE(M)
ACTIVATE(n__0) → 0¹
U15¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U12¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U64¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U14¹(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → S(activate(X))
U61¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V2) → U32¹(isNatKind(activate(V2)))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U61¹(tt, M, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U52¹(tt, N) → ACTIVATE(N)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U31¹(tt, V2) → ISNATKIND(activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, M, N) → ISNATKIND(activate(M))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U22¹(tt, V1) → U23¹(isNat(activate(V1)))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U64¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → S(plus(activate(N), activate(M)))
U63¹(tt, M, N) → ACTIVATE(N)
PLUS(N, s(M)) → ISNAT(M)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
U51¹(tt, N) → ACTIVATE(N)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U12¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
PLUS(N, 0) → ISNAT(N)
U14¹(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, M, N) → ISNAT(activate(N))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U22¹(tt, V1) → ACTIVATE(V1)
U13¹(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U62¹(tt, M, N) → ACTIVATE(N)
U15¹(tt, V2) → U16¹(isNat(activate(V2)))
U21¹(tt, V1) → ACTIVATE(V1)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U63¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U13¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U51¹(isNat(N), N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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 7 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

U11¹(tt, V1, V2) → ACTIVATE(V2)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U62¹(tt, M, N) → ACTIVATE(M)
U15¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U64¹(tt, M, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
U14¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U61¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U52¹(tt, N) → ACTIVATE(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U31¹(tt, V2) → ISNATKIND(activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, M, N) → ISNATKIND(activate(M))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U64¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → ACTIVATE(N)
PLUS(N, s(M)) → ISNAT(M)
U51¹(tt, N) → ACTIVATE(N)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
U12¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
PLUS(N, 0) → ISNAT(N)
U14¹(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, M, N) → ISNAT(activate(N))
U22¹(tt, V1) → ISNAT(activate(V1))
U13¹(tt, V1, V2) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
U22¹(tt, V1) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
U62¹(tt, M, N) → ACTIVATE(N)
U21¹(tt, V1) → ACTIVATE(V1)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U63¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U13¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U51¹(isNat(N), N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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.

ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.

U11¹(tt, V1, V2) → ACTIVATE(V2)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U62¹(tt, M, N) → ACTIVATE(M)
U15¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U64¹(tt, M, N) → ACTIVATE(N)
U14¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U61¹(tt, M, N) → ACTIVATE(N)
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U52¹(tt, N) → ACTIVATE(N)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U31¹(tt, V2) → ISNATKIND(activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
U61¹(tt, M, N) → ISNATKIND(activate(M))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U64¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → ACTIVATE(N)
PLUS(N, s(M)) → ISNAT(M)
U51¹(tt, N) → ACTIVATE(N)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
PLUS(N, 0) → ISNAT(N)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U62¹(tt, M, N) → ISNAT(activate(N))
U22¹(tt, V1) → ISNAT(activate(V1))
U13¹(tt, V1, V2) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U62¹(tt, M, N) → ACTIVATE(N)
U21¹(tt, V1) → ACTIVATE(V1)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U63¹(tt, M, N) → ACTIVATE(M)
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U13¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U51¹(isNat(N), N)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(PLUS(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂, x₃)) = x₂ + x₃
POL(U11¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U12(x₁, x₂, x₃)) = x₂ + x₃
POL(U12¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U13(x₁, x₂, x₃)) = x₂ + x₃
POL(U13¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U14(x₁, x₂, x₃)) = x₂ + x₃
POL(U14¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U15(x₁, x₂)) = 0
POL(U15¹(x₁, x₂)) = x₂
POL(U16(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21¹(x₁, x₂)) = x₂
POL(U22(x₁, x₂)) = 0
POL(U22¹(x₁, x₂)) = x₂
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U31¹(x₁, x₂)) = x₂
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(U51(x₁, x₂)) = 1 + x₂
POL(U51¹(x₁, x₂)) = x₂
POL(U52(x₁, x₂)) = 1 + x₂
POL(U52¹(x₁, x₂)) = x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U61¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U62¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U63¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64¹(x₁, x₂, x₃)) = x₂ + x₃
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = 1 + x₁ + x₂
POL(n__s(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

The following usable rules [17] were oriented:

U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U16(tt) → tt
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U41(tt) → tt
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__0) → 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)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U62¹(tt, M, N) → ACTIVATE(M)
U15¹(tt, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U64¹(tt, M, N) → ACTIVATE(N)
U14¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V1)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U61¹(tt, M, N) → ACTIVATE(N)
U52¹(tt, N) → ACTIVATE(N)
U31¹(tt, V2) → ISNATKIND(activate(V2))
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U15¹(tt, V2) → ISNAT(activate(V2))
U61¹(tt, M, N) → ISNATKIND(activate(M))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U64¹(tt, M, N) → ACTIVATE(M)
U63¹(tt, M, N) → ACTIVATE(N)
PLUS(N, s(M)) → ISNAT(M)
U51¹(tt, N) → ACTIVATE(N)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U31¹(tt, V2) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, N) → ISNATKIND(activate(N))
U63¹(tt, M, N) → ISNATKIND(activate(N))
PLUS(N, 0) → ISNAT(N)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U62¹(tt, M, N) → ISNAT(activate(N))
U22¹(tt, V1) → ISNAT(activate(V1))
U13¹(tt, V1, V2) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U62¹(tt, M, N) → ACTIVATE(N)
U21¹(tt, V1) → ACTIVATE(V1)
U63¹(tt, M, N) → ACTIVATE(M)
U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U51¹(tt, N) → U52¹(isNatKind(activate(N)), activate(N))
U13¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, 0) → U51¹(isNat(N), N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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 4 SCCs with 44 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

ACTIVATE(n__s(X)) → ACTIVATE(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ACTIVATE(n__s(X)) → ACTIVATE(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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__s(V1)) → ISNATKIND(activate(V1))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂, x₃)) = 0
POL(U13(x₁, x₂, x₃)) = 0
POL(U14(x₁, x₂, x₃)) = 0
POL(U15(x₁, x₂)) = 0
POL(U16(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(U51(x₁, x₂)) = x₂
POL(U52(x₁, x₂)) = x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [17] were oriented:

U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U16(tt) → tt
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U41(tt) → tt
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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)) → U21¹(isNatKind(activate(V1)), activate(V1))

The remaining pairs can at least be oriented weakly.

U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂, x₃)) = 0
POL(U13(x₁, x₂, x₃)) = 0
POL(U14(x₁, x₂, x₃)) = 0
POL(U15(x₁, x₂)) = 0
POL(U16(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21¹(x₁, x₂)) = x₂
POL(U22(x₁, x₂)) = 0
POL(U22¹(x₁, x₂)) = max(x₁, x₂)
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(U51(x₁, x₂)) = x₂
POL(U52(x₁, x₂)) = x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [17] were oriented:

U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U16(tt) → tt
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U41(tt) → tt
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                  ↳ QDP

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

U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

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

U64¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → U61¹(isNat(M), M, N)
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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.

PLUS(N, s(M)) → U61¹(isNat(M), M, N)

The remaining pairs can at least be oriented weakly.

U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1
POL(PLUS(x₁, x₂)) = x₂
POL(U11(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂, x₃)) = 0
POL(U13(x₁, x₂, x₃)) = 0
POL(U14(x₁, x₂, x₃)) = 0
POL(U15(x₁, x₂)) = 0
POL(U16(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁)) = 0
POL(U51(x₁, x₂)) = 1 + x₂
POL(U52(x₁, x₂)) = x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U61¹(x₁, x₂, x₃)) = x₂
POL(U62(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U62¹(x₁, x₂, x₃)) = x₂
POL(U63(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U63¹(x₁, x₂, x₃)) = x₂
POL(U64(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U64¹(x₁, x₂, x₃)) = x₂
POL(activate(x₁)) = x₁
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(n__0) = 1
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [17] were oriented:

U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U16(tt) → tt
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U32(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U41(tt) → tt
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
plus(N, 0) → U51(isNat(N), N)
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

U64¹(tt, M, N) → PLUS(activate(N), activate(M))
U63¹(tt, M, N) → U64¹(isNatKind(activate(N)), activate(M), activate(N))
U62¹(tt, M, N) → U63¹(isNat(activate(N)), activate(M), activate(N))
U61¹(tt, M, N) → U62¹(isNatKind(activate(M)), activate(M), activate(N))

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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.