Termination w.r.t. Q proof of /tmp/tmppT2FkL/MYNAT_complete-noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

(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(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Q is empty.

(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(activate(X1), activate(X2))
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(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(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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

(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(activate(X1), activate(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__plus(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__plus(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
X(N, 0) → U91¹(isNat(N), N)
U91¹(tt, N) → ISNATKIND(activate(N))
ISNATKIND(n__plus(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__x(X1, X2)) → ACTIVATE(X2)
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(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

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