Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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

U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0
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))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

U101¹(tt, M, N) → U102¹(isNatKind(activate(M)), activate(M), activate(N))
U101¹(tt, M, N) → ISNATKIND(activate(M))
U101¹(tt, M, N) → ACTIVATE(M)
U101¹(tt, M, N) → ACTIVATE(N)
U102¹(tt, M, N) → U103¹(isNat(activate(N)), activate(M), activate(N))
U102¹(tt, M, N) → ISNAT(activate(N))
U102¹(tt, M, N) → ACTIVATE(N)
U102¹(tt, M, N) → ACTIVATE(M)
U103¹(tt, M, N) → U104¹(isNatKind(activate(N)), activate(M), activate(N))
U103¹(tt, M, N) → ISNATKIND(activate(N))
U103¹(tt, M, N) → ACTIVATE(N)
U103¹(tt, M, N) → ACTIVATE(M)
U104¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U104¹(tt, M, N) → X(activate(N), activate(M))
U104¹(tt, M, N) → ACTIVATE(N)
U104¹(tt, M, N) → ACTIVATE(M)
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U15¹(tt, V2) → U16¹(isNat(activate(V2)))
U15¹(tt, V2) → ISNAT(activate(V2))
U15¹(tt, V2) → ACTIVATE(V2)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U22¹(tt, V1) → U23¹(isNat(activate(V1)))
U22¹(tt, V1) → ISNAT(activate(V1))
U22¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, V1, V2) → U32¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U31¹(tt, V1, V2) → ISNATKIND(activate(V1))
U31¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ACTIVATE(V2)
U32¹(tt, V1, V2) → U33¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U32¹(tt, V1, V2) → ISNATKIND(activate(V2))
U32¹(tt, V1, V2) → ACTIVATE(V2)
U32¹(tt, V1, V2) → ACTIVATE(V1)
U33¹(tt, V1, V2) → U34¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U33¹(tt, V1, V2) → ISNATKIND(activate(V2))
U33¹(tt, V1, V2) → ACTIVATE(V2)
U33¹(tt, V1, V2) → ACTIVATE(V1)
U34¹(tt, V1, V2) → U35¹(isNat(activate(V1)), activate(V2))
U34¹(tt, V1, V2) → ISNAT(activate(V1))
U34¹(tt, V1, V2) → ACTIVATE(V1)
U34¹(tt, V1, V2) → ACTIVATE(V2)
U35¹(tt, V2) → U36¹(isNat(activate(V2)))
U35¹(tt, V2) → ISNAT(activate(V2))
U35¹(tt, V2) → ACTIVATE(V2)
U41¹(tt, V2) → U42¹(isNatKind(activate(V2)))
U41¹(tt, V2) → ISNATKIND(activate(V2))
U41¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V2) → U62¹(isNatKind(activate(V2)))
U61¹(tt, V2) → ISNATKIND(activate(V2))
U61¹(tt, V2) → ACTIVATE(V2)
U71¹(tt, N) → U72¹(isNatKind(activate(N)), activate(N))
U71¹(tt, N) → ISNATKIND(activate(N))
U71¹(tt, N) → ACTIVATE(N)
U72¹(tt, N) → ACTIVATE(N)
U81¹(tt, M, N) → U82¹(isNatKind(activate(M)), activate(M), activate(N))
U81¹(tt, M, N) → ISNATKIND(activate(M))
U81¹(tt, M, N) → ACTIVATE(M)
U81¹(tt, M, N) → ACTIVATE(N)
U82¹(tt, M, N) → U83¹(isNat(activate(N)), activate(M), activate(N))
U82¹(tt, M, N) → ISNAT(activate(N))
U82¹(tt, M, N) → ACTIVATE(N)
U82¹(tt, M, N) → ACTIVATE(M)
U83¹(tt, M, N) → U84¹(isNatKind(activate(N)), activate(M), activate(N))
U83¹(tt, M, N) → ISNATKIND(activate(N))
U83¹(tt, M, N) → ACTIVATE(N)
U83¹(tt, M, N) → ACTIVATE(M)
U84¹(tt, M, N) → S(plus(activate(N), activate(M)))
U84¹(tt, M, N) → PLUS(activate(N), activate(M))
U84¹(tt, M, N) → ACTIVATE(N)
U84¹(tt, M, N) → ACTIVATE(M)
U91¹(tt, N) → U92¹(isNatKind(activate(N)))
U91¹(tt, N) → ISNATKIND(activate(N))
U91¹(tt, N) → ACTIVATE(N)
U92¹(tt) → 0¹
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__plus(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → U51¹(isNatKind(activate(V1)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
PLUS(N, 0) → U71¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U81¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U91¹(isNat(N), N)
X(N, 0) → ISNAT(N)
X(N, s(M)) → U101¹(isNat(M), M, N)
X(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0
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))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes.

(4) Obligation:

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

U102¹(tt, M, N) → U103¹(isNat(activate(N)), activate(M), activate(N))
U103¹(tt, M, N) → U104¹(isNatKind(activate(N)), activate(M), activate(N))
U104¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → U71¹(isNat(N), N)
U71¹(tt, N) → U72¹(isNatKind(activate(N)), activate(N))
U72¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V2))
U41¹(tt, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
X(N, 0) → U91¹(isNat(N), N)
U91¹(tt, N) → ISNATKIND(activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
U91¹(tt, N) → ACTIVATE(N)
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U31¹(tt, V1, V2) → U32¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U32¹(tt, V1, V2) → U33¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U33¹(tt, V1, V2) → U34¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U34¹(tt, V1, V2) → U35¹(isNat(activate(V1)), activate(V2))
U35¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U35¹(tt, V2) → ACTIVATE(V2)
U34¹(tt, V1, V2) → ISNAT(activate(V1))
U34¹(tt, V1, V2) → ACTIVATE(V1)
U34¹(tt, V1, V2) → ACTIVATE(V2)
U33¹(tt, V1, V2) → ISNATKIND(activate(V2))
U33¹(tt, V1, V2) → ACTIVATE(V2)
U33¹(tt, V1, V2) → ACTIVATE(V1)
U32¹(tt, V1, V2) → ISNATKIND(activate(V2))
U32¹(tt, V1, V2) → ACTIVATE(V2)
U32¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ISNATKIND(activate(V1))
U31¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ACTIVATE(V2)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
X(N, s(M)) → U101¹(isNat(M), M, N)
U101¹(tt, M, N) → U102¹(isNatKind(activate(M)), activate(M), activate(N))
U102¹(tt, M, N) → ISNAT(activate(N))
U102¹(tt, M, N) → ACTIVATE(N)
U102¹(tt, M, N) → ACTIVATE(M)
U101¹(tt, M, N) → ISNATKIND(activate(M))
U101¹(tt, M, N) → ACTIVATE(M)
U101¹(tt, M, N) → ACTIVATE(N)
X(N, s(M)) → ISNAT(M)
U41¹(tt, V2) → ACTIVATE(V2)
U15¹(tt, V2) → ACTIVATE(V2)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U81¹(isNat(M), M, N)
U81¹(tt, M, N) → U82¹(isNatKind(activate(M)), activate(M), activate(N))
U82¹(tt, M, N) → U83¹(isNat(activate(N)), activate(M), activate(N))
U83¹(tt, M, N) → U84¹(isNatKind(activate(N)), activate(M), activate(N))
U84¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U84¹(tt, M, N) → ACTIVATE(N)
U84¹(tt, M, N) → ACTIVATE(M)
U83¹(tt, M, N) → ISNATKIND(activate(N))
U83¹(tt, M, N) → ACTIVATE(N)
U83¹(tt, M, N) → ACTIVATE(M)
U82¹(tt, M, N) → ISNAT(activate(N))
U82¹(tt, M, N) → ACTIVATE(N)
U82¹(tt, M, N) → ACTIVATE(M)
U81¹(tt, M, N) → ISNATKIND(activate(M))
U81¹(tt, M, N) → ACTIVATE(M)
U81¹(tt, M, N) → ACTIVATE(N)
U71¹(tt, N) → ISNATKIND(activate(N))
U71¹(tt, N) → ACTIVATE(N)
U104¹(tt, M, N) → X(activate(N), activate(M))
U104¹(tt, M, N) → ACTIVATE(N)
U104¹(tt, M, N) → ACTIVATE(M)
U103¹(tt, M, N) → ISNATKIND(activate(N))
U103¹(tt, M, N) → ACTIVATE(N)
U103¹(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0
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))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
PLUS(N, 0) → U71¹(isNat(N), N)
U71¹(tt, N) → U72¹(isNatKind(activate(N)), activate(N))
U72¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → U11¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V2))
ISNATKIND(n__plus(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V1)
X(N, 0) → U91¹(isNat(N), N)
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATKIND(activate(V2))
ISNATKIND(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V1)
ISNATKIND(n__x(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
U91¹(tt, N) → ACTIVATE(N)
X(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → U31¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U31¹(tt, V1, V2) → U32¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U32¹(tt, V1, V2) → U33¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U34¹(tt, V1, V2) → U35¹(isNat(activate(V1)), activate(V2))
U35¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__x(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__x(V1, V2)) → ACTIVATE(V1)
ISNAT(n__x(V1, V2)) → ACTIVATE(V2)
U35¹(tt, V2) → ACTIVATE(V2)
U34¹(tt, V1, V2) → ISNAT(activate(V1))
U34¹(tt, V1, V2) → ACTIVATE(V1)
U34¹(tt, V1, V2) → ACTIVATE(V2)
U33¹(tt, V1, V2) → ISNATKIND(activate(V2))
U33¹(tt, V1, V2) → ACTIVATE(V2)
U33¹(tt, V1, V2) → ACTIVATE(V1)
U32¹(tt, V1, V2) → ISNATKIND(activate(V2))
U32¹(tt, V1, V2) → ACTIVATE(V2)
U32¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ISNATKIND(activate(V1))
U31¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V1, V2) → ACTIVATE(V2)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
X(N, s(M)) → U101¹(isNat(M), M, N)
U102¹(tt, M, N) → ISNAT(activate(N))
U102¹(tt, M, N) → ACTIVATE(N)
U102¹(tt, M, N) → ACTIVATE(M)
U101¹(tt, M, N) → ISNATKIND(activate(M))
U101¹(tt, M, N) → ACTIVATE(M)
U101¹(tt, M, N) → ACTIVATE(N)
X(N, s(M)) → ISNAT(M)
U41¹(tt, V2) → ACTIVATE(V2)
U15¹(tt, V2) → ACTIVATE(V2)
U14¹(tt, V1, V2) → ISNAT(activate(V1))
U14¹(tt, V1, V2) → ACTIVATE(V1)
U14¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ISNATKIND(activate(V2))
U13¹(tt, V1, V2) → ACTIVATE(V2)
U13¹(tt, V1, V2) → ACTIVATE(V1)
U12¹(tt, V1, V2) → ISNATKIND(activate(V2))
U12¹(tt, V1, V2) → ACTIVATE(V2)
U12¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ISNATKIND(activate(V1))
U11¹(tt, V1, V2) → ACTIVATE(V1)
U11¹(tt, V1, V2) → ACTIVATE(V2)
PLUS(N, s(M)) → U81¹(isNat(M), M, N)
U84¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U84¹(tt, M, N) → ACTIVATE(N)
U84¹(tt, M, N) → ACTIVATE(M)
U83¹(tt, M, N) → ISNATKIND(activate(N))
U83¹(tt, M, N) → ACTIVATE(N)
U83¹(tt, M, N) → ACTIVATE(M)
U82¹(tt, M, N) → ISNAT(activate(N))
U82¹(tt, M, N) → ACTIVATE(N)
U82¹(tt, M, N) → ACTIVATE(M)
U81¹(tt, M, N) → ISNATKIND(activate(M))
U81¹(tt, M, N) → ACTIVATE(M)
U81¹(tt, M, N) → ACTIVATE(N)
U71¹(tt, N) → ISNATKIND(activate(N))
U71¹(tt, N) → ACTIVATE(N)
U104¹(tt, M, N) → X(activate(N), activate(M))
U104¹(tt, M, N) → ACTIVATE(N)
U104¹(tt, M, N) → ACTIVATE(M)
U103¹(tt, M, N) → ISNATKIND(activate(N))
U103¹(tt, M, N) → ACTIVATE(N)
U103¹(tt, M, N) → ACTIVATE(M)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  U102¹(x1, x2, x3)
tt  =  tt
U103¹(x1, x2, x3)  =  U103¹(x1, x2, x3)
isNat(x1)  =  isNat
activate(x1)  =  x1
U104¹(x1, x2, x3)  =  U104¹(x1, x2, x3)
isNatKind(x1)  =  isNatKind
PLUS(x1, x2)  =  PLUS(x1, x2)
x(x1, x2)  =  x(x1, x2)
0  =  0
U71¹(x1, x2)  =  U71¹(x2)
U72¹(x1, x2)  =  U72¹(x2)
ACTIVATE(x1)  =  x1
n__plus(x1, x2)  =  n__plus(x1, x2)
ISNAT(x1)  =  ISNAT(x1)
U11¹(x1, x2, x3)  =  U11¹(x1, x2, x3)
U12¹(x1, x2, x3)  =  U12¹(x1, x2, x3)
U13¹(x1, x2, x3)  =  U13¹(x2, x3)
U14¹(x1, x2, x3)  =  U14¹(x2, x3)
U15¹(x1, x2)  =  U15¹(x2)
ISNATKIND(x1)  =  ISNATKIND(x1)
U41¹(x1, x2)  =  U41¹(x2)
n__x(x1, x2)  =  n__x(x1, x2)
X(x1, x2)  =  X(x1, x2)
U91¹(x1, x2)  =  U91¹(x2)
n__s(x1)  =  n__s(x1)
U61¹(x1, x2)  =  U61¹(x1, x2)
U21¹(x1, x2)  =  U21¹(x1, x2)
U22¹(x1, x2)  =  U22¹(x1, x2)
U31¹(x1, x2, x3)  =  U31¹(x1, x2, x3)
U32¹(x1, x2, x3)  =  U32¹(x1, x2, x3)
U33¹(x1, x2, x3)  =  U33¹(x1, x2, x3)
U34¹(x1, x2, x3)  =  U34¹(x1, x2, x3)
U35¹(x1, x2)  =  U35¹(x1, x2)
s(x1)  =  s(x1)
U101¹(x1, x2, x3)  =  U101¹(x1, x2, x3)
U81¹(x1, x2, x3)  =  U81¹(x1, x2, x3)
U82¹(x1, x2, x3)  =  U82¹(x1, x2, x3)
U83¹(x1, x2, x3)  =  U83¹(x1, x2, x3)
U84¹(x1, x2, x3)  =  U84¹(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
n__0  =  n__0
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U91(x1, x2)  =  U91(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
U72(x1, x2)  =  U72(x1, x2)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
U71(x1, x2)  =  U71(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14
U15(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23
U21(x1, x2)  =  x1
U16(x1)  =  U16
U33(x1, x2, x3)  =  U33
U34(x1, x2, x3)  =  U34
U32(x1, x2, x3)  =  U32
U31(x1, x2, x3)  =  x1
U41(x1, x2)  =  U41
U42(x1)  =  x1
U36(x1)  =  U36
U35(x1, x2)  =  U35
U62(x1)  =  x1
U61(x1, x2)  =  U61
U51(x1)  =  U51
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U92(x1)  =  U92(x1)
U84(x1, x2, x3)  =  U84(x1, x2, x3)

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

[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > [PLUS₂, U81^1₃, U82^1₃, U83^1₃, U84^1₃] > [0, U71^1₁, n₀] > U72^1₁
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > [PLUS₂, U81^1₃, U82^1₃, U83^1₃, U84^1₃] > [0, U71^1₁, n₀] > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U14^1₂ > U15^1₁ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U21^1₂ > U22^1₂ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U32^1₃ > [U33^1₃, U34^1₃] > U35^1₂ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U72₂
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [n_plus2, plus₂, U81₃, U82₃, U83₃, U84₃] > U71₂ > U72₂
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > U61^1₂ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > U31^1₃ > U32^1₃ > [U33^1₃, U34^1₃] > U35^1₂ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [U91₁, U92₁] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > [PLUS₂, U81^1₃, U82^1₃, U83^1₃, U84^1₃] > [0, U71^1₁, n₀] > U72^1₁
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [U91₁, U92₁] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > [PLUS₂, U81^1₃, U82^1₃, U83^1₃, U84^1₃] > [0, U71^1₁, n₀] > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [U91₁, U92₁] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U14^1₂ > U15^1₁ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [U91₁, U92₁] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U21^1₂ > U22^1₂ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [U91₁, U92₁] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U32^1₃ > [U33^1₃, U34^1₃] > U35^1₂ > ISNAT₁ > [ISNATKIND₁, U41^1₁, U91^1₁]
[U102^1₃, U103^1₃, U104^1₃, x₂, n_x2, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [U91₁, U92₁] > [tt, isNat, isNatKind, U11^1₃, U12^1₃, U13^1₂, n_s1, s₁, U14, U23, U16, U33, U34, U32, U41, U36, U35, U61, U51] > U72₂

Status:

n_plus2: multiset
U35: []
U41^1₁: [1]
U83^1₃: multiset
U34: []
U84^1₃: multiset
U33: []
n_s1: [1]
U103₃: multiset
U71₂: multiset
U72^1₁: multiset
U101₃: multiset
tt: multiset
U102^1₃: multiset
U41: []
U14^1₂: multiset
isNatKind: []
U82^1₃: multiset
plus₂: multiset
X₂: multiset
U51: []
U91^1₁: [1]
U31^1₃: [2,1,3]
U15^1₁: [1]
U102₃: multiset
ISNAT₁: [1]
U36: []
U33^1₃: multiset
U32^1₃: multiset
U71^1₁: multiset
n_x2: multiset
U101^1₃: multiset
U103^1₃: multiset
U82₃: multiset
U22^1₂: multiset
U61^1₂: multiset
U21^1₂: [2,1]
x₂: multiset
U104₃: multiset
isNat: []
U12^1₃: multiset
PLUS₂: multiset
U61: []
U91₁: multiset
U81^1₃: multiset
U92₁: multiset
U14: []
s₁: [1]
U23: []
U83₃: multiset
U32: []
ISNATKIND₁: [1]
U11^1₃: multiset
U35^1₂: multiset
U81₃: multiset
U104^1₃: multiset
U72₂: [1,2]
0: multiset
U16: []
U34^1₃: multiset
n₀: multiset
U13^1₂: multiset
U84₃: multiset

The following usable rules [FROCOS05] were oriented:

plus(X1, X2) → n__plus(X1, X2)
0 → n__0
x(X1, X2) → n__x(X1, X2)
s(X) → n__s(X)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
activate(X) → X
activate(n__0) → 0
activate(n__s(X)) → s(X)
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U72(tt, N) → activate(N)
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
x(N, s(M)) → U101(isNat(M), M, N)
activate(n__plus(X1, X2)) → plus(X1, X2)
plus(N, 0) → U71(isNat(N), N)
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
activate(n__x(X1, X2)) → x(X1, X2)
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))
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U16(tt) → tt
U15(tt, V2) → U16(isNat(activate(V2)))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U23(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U36(tt) → tt
U35(tt, V2) → U36(isNat(activate(V2)))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U62(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U51(tt) → tt
U42(tt) → tt
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))

(6) Obligation:

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

U102¹(tt, M, N) → U103¹(isNat(activate(N)), activate(M), activate(N))
U103¹(tt, M, N) → U104¹(isNatKind(activate(N)), activate(M), activate(N))
U11¹(tt, V1, V2) → U12¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V2) → ISNATKIND(activate(V2))
ACTIVATE(n__x(X1, X2)) → X(X1, X2)
U91¹(tt, N) → ISNATKIND(activate(N))
U33¹(tt, V1, V2) → U34¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U101¹(tt, M, N) → U102¹(isNatKind(activate(M)), activate(M), activate(N))
U81¹(tt, M, N) → U82¹(isNatKind(activate(M)), activate(M), activate(N))
U82¹(tt, M, N) → U83¹(isNat(activate(N)), activate(M), activate(N))
U83¹(tt, M, N) → U84¹(isNatKind(activate(N)), activate(M), activate(N))

The TRS R consists of the following rules:

U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
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, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0
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))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 11 less nodes.

(8) TRUE