Termination w.r.t. Q proof of /tmp/tmp0eER7j/MYNAT_complete-noand

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 TRUE

(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.

(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)
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))
U14¹(tt, V1, V2) → U15¹(isNat(activate(V1)), 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)
X(N, 0) → U91¹(isNat(N), 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)
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))
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)
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)
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)
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) = x2
U72¹(x1, x2) = x2
ACTIVATE(x1) = x1
n__plus(x1, x2) = n__plus(x1, x2)
ISNAT(x1) = x1
U11¹(x1, x2, x3) = U11¹(x2, x3)
U12¹(x1, x2, x3) = U12¹(x2, x3)
U13¹(x1, x2, x3) = U13¹(x2, x3)
U14¹(x1, x2, x3) = U14¹(x2, x3)
U15¹(x1, x2) = x2
ISNATKIND(x1) = x1
U41¹(x1, x2) = U41¹(x1, x2)
n__s(x1) = n__s(x1)
n__x(x1, x2) = n__x(x1, x2)
X(x1, x2) = X(x1, x2)
U91¹(x1, x2) = x2
U61¹(x1, x2) = U61¹(x1, x2)
U21¹(x1, x2) = x2
U22¹(x1, x2) = 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)
U101(x1, x2, x3) = U101(x1, x2, x3)
U102(x1, x2, x3) = U102(x1, x2, x3)
U103(x1, x2, x3) = U103(x1, x2, x3)
U104(x1, x2, x3) = U104(x1, x2, x3)
plus(x1, x2) = plus(x1, x2)
U11(x1, x2, x3) = U11
U12(x1, x2, x3) = x1
U13(x1, x2, x3) = U13
U14(x1, x2, x3) = U14
U15(x1, x2) = U15
U16(x1) = x1
U21(x1, x2) = U21
U22(x1, x2) = x1
U23(x1) = x1
U31(x1, x2, x3) = x1
U32(x1, x2, x3) = x1
U33(x1, x2, x3) = U33
U34(x1, x2, x3) = U34
U35(x1, x2) = x1
U36(x1) = U36
U41(x1, x2) = U41
U42(x1) = x1
U51(x1) = x1
U61(x1, x2) = x1
U62(x1) = x1
U71(x1, x2) = x2
U72(x1, x2) = x2
U81(x1, x2, x3) = U81(x1, x2, x3)
U82(x1, x2, x3) = U82(x1, x2, x3)
U83(x1, x2, x3) = U83(x1, x2, x3)
U84(x1, x2, x3) = U84(x1, x2, x3)
U91(x1, x2) = x1
U92(x1) = U92
n__0 = n__0

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U102^1₃, U103^1₃, U104^1₃, x₂, n_{_x₂}, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [tt, isNat, isNatKind, PLUS₂, 0, n_{_plus₂}, U81^1₃, U82^1₃, U83^1₃, U84^1₃, plus₂, U11, U13, U14, U15, U21, U33, U34, U36, U41, U81₃, U82₃, U83₃, U84₃, U92, n_₀] > U11^1₂ > [U12^1₂, U13^1₂, U14^1₂] > U35^1₂
[U102^1₃, U103^1₃, U104^1₃, x₂, n_{_x₂}, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [tt, isNat, isNatKind, PLUS₂, 0, n_{_plus₂}, U81^1₃, U82^1₃, U83^1₃, U84^1₃, plus₂, U11, U13, U14, U15, U21, U33, U34, U36, U41, U81₃, U82₃, U83₃, U84₃, U92, n_₀] > U41^1₂ > U35^1₂
[U102^1₃, U103^1₃, U104^1₃, x₂, n_{_x₂}, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [tt, isNat, isNatKind, PLUS₂, 0, n_{_plus₂}, U81^1₃, U82^1₃, U83^1₃, U84^1₃, plus₂, U11, U13, U14, U15, U21, U33, U34, U36, U41, U81₃, U82₃, U83₃, U84₃, U92, n_₀] > [n_{_s₁}, s₁] > U35^1₂
[U102^1₃, U103^1₃, U104^1₃, x₂, n_{_x₂}, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > [tt, isNat, isNatKind, PLUS₂, 0, n_{_plus₂}, U81^1₃, U82^1₃, U83^1₃, U84^1₃, plus₂, U11, U13, U14, U15, U21, U33, U34, U36, U41, U81₃, U82₃, U83₃, U84₃, U92, n_₀] > U32^1₃ > U33^1₃ > U34^1₃ > U35^1₂
[U102^1₃, U103^1₃, U104^1₃, x₂, n_{_x₂}, X₂, U101^1₃, U101₃, U102₃, U103₃, U104₃] > U61^1₂ > U35^1₂
[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

Status:

U102^1₃: [2,3,1]
tt: []
U103^1₃: [2,3,1]
isNat: []
U104^1₃: [2,3,1]
isNatKind: []
PLUS₂: [1,2]
x₂: [2,1]
0: []
n_{_plus₂}: [1,2]
U11^1₂: [1,2]
U12^1₂: [2,1]
U13^1₂: [2,1]
U14^1₂: [2,1]
U41^1₂: [2,1]
n_{_s₁}: [1]
n_{_x₂}: [2,1]
X₂: [2,1]
U61^1₂: [1,2]
U31^1₃: [2,3,1]
U32^1₃: [2,1,3]
U33^1₃: [3,2,1]
U34^1₃: [3,2,1]
U35^1₂: [2,1]
s₁: [1]
U101^1₃: [2,3,1]
U81^1₃: [3,2,1]
U82^1₃: [3,2,1]
U83^1₃: [3,2,1]
U84^1₃: [3,2,1]
U101₃: [2,3,1]
U102₃: [2,3,1]
U103₃: [2,3,1]
U104₃: [2,3,1]
plus₂: [1,2]
U11: []
U13: []
U14: []
U15: []
U21: []
U33: []
U34: []
U36: []
U41: []
U81₃: [3,2,1]
U82₃: [3,2,1]
U83₃: [3,2,1]
U84₃: [3,2,1]
U92: []
n_₀: []

The following usable rules [FROCOS05] were oriented:

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

(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))
U71¹(tt, N) → U72¹(isNatKind(activate(N)), activate(N))
U72¹(tt, N) → ACTIVATE(N)
ACTIVATE(n__plus(X1, X2)) → PLUS(activate(X1), activate(X2))
U12¹(tt, V1, V2) → U13¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U13¹(tt, V1, V2) → U14¹(isNatKind(activate(V2)), activate(V1), activate(V2))
U15¹(tt, V2) → ISNAT(activate(V2))
ACTIVATE(n__x(X1, X2)) → X(activate(X1), activate(X2))
U91¹(tt, N) → ISNATKIND(activate(N))
U91¹(tt, N) → ACTIVATE(N)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U101¹(tt, M, N) → U102¹(isNatKind(activate(M)), activate(M), activate(N))
U15¹(tt, V2) → ACTIVATE(V2)
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))
U71¹(tt, N) → ISNATKIND(activate(N))
U71¹(tt, N) → 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(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.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 23 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) TRUE