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

(0) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(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:

ACTIVE(U101(tt, M, N)) → MARK(U102(isNatKind(M), M, N))
ACTIVE(U101(tt, M, N)) → U102¹(isNatKind(M), M, N)
ACTIVE(U101(tt, M, N)) → ISNATKIND(M)
ACTIVE(U102(tt, M, N)) → MARK(U103(isNat(N), M, N))
ACTIVE(U102(tt, M, N)) → U103¹(isNat(N), M, N)
ACTIVE(U102(tt, M, N)) → ISNAT(N)
ACTIVE(U103(tt, M, N)) → MARK(U104(isNatKind(N), M, N))
ACTIVE(U103(tt, M, N)) → U104¹(isNatKind(N), M, N)
ACTIVE(U103(tt, M, N)) → ISNATKIND(N)
ACTIVE(U104(tt, M, N)) → MARK(plus(x(N, M), N))
ACTIVE(U104(tt, M, N)) → PLUS(x(N, M), N)
ACTIVE(U104(tt, M, N)) → X(N, M)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
ACTIVE(U11(tt, V1, V2)) → U12¹(isNatKind(V1), V1, V2)
ACTIVE(U11(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U12(tt, V1, V2)) → U13¹(isNatKind(V2), V1, V2)
ACTIVE(U12(tt, V1, V2)) → ISNATKIND(V2)
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U13(tt, V1, V2)) → U14¹(isNatKind(V2), V1, V2)
ACTIVE(U13(tt, V1, V2)) → ISNATKIND(V2)
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U14(tt, V1, V2)) → U15¹(isNat(V1), V2)
ACTIVE(U14(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U15(tt, V2)) → U16¹(isNat(V2))
ACTIVE(U15(tt, V2)) → ISNAT(V2)
ACTIVE(U16(tt)) → MARK(tt)
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U21(tt, V1)) → U22¹(isNatKind(V1), V1)
ACTIVE(U21(tt, V1)) → ISNATKIND(V1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U22(tt, V1)) → U23¹(isNat(V1))
ACTIVE(U22(tt, V1)) → ISNAT(V1)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNatKind(V1), V1, V2))
ACTIVE(U31(tt, V1, V2)) → U32¹(isNatKind(V1), V1, V2)
ACTIVE(U31(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U32(tt, V1, V2)) → MARK(U33(isNatKind(V2), V1, V2))
ACTIVE(U32(tt, V1, V2)) → U33¹(isNatKind(V2), V1, V2)
ACTIVE(U32(tt, V1, V2)) → ISNATKIND(V2)
ACTIVE(U33(tt, V1, V2)) → MARK(U34(isNatKind(V2), V1, V2))
ACTIVE(U33(tt, V1, V2)) → U34¹(isNatKind(V2), V1, V2)
ACTIVE(U33(tt, V1, V2)) → ISNATKIND(V2)
ACTIVE(U34(tt, V1, V2)) → MARK(U35(isNat(V1), V2))
ACTIVE(U34(tt, V1, V2)) → U35¹(isNat(V1), V2)
ACTIVE(U34(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U35(tt, V2)) → MARK(U36(isNat(V2)))
ACTIVE(U35(tt, V2)) → U36¹(isNat(V2))
ACTIVE(U35(tt, V2)) → ISNAT(V2)
ACTIVE(U36(tt)) → MARK(tt)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatKind(V2)))
ACTIVE(U41(tt, V2)) → U42¹(isNatKind(V2))
ACTIVE(U41(tt, V2)) → ISNATKIND(V2)
ACTIVE(U42(tt)) → MARK(tt)
ACTIVE(U51(tt)) → MARK(tt)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatKind(V2)))
ACTIVE(U61(tt, V2)) → U62¹(isNatKind(V2))
ACTIVE(U61(tt, V2)) → ISNATKIND(V2)
ACTIVE(U62(tt)) → MARK(tt)
ACTIVE(U71(tt, N)) → MARK(U72(isNatKind(N), N))
ACTIVE(U71(tt, N)) → U72¹(isNatKind(N), N)
ACTIVE(U71(tt, N)) → ISNATKIND(N)
ACTIVE(U72(tt, N)) → MARK(N)
ACTIVE(U81(tt, M, N)) → MARK(U82(isNatKind(M), M, N))
ACTIVE(U81(tt, M, N)) → U82¹(isNatKind(M), M, N)
ACTIVE(U81(tt, M, N)) → ISNATKIND(M)
ACTIVE(U82(tt, M, N)) → MARK(U83(isNat(N), M, N))
ACTIVE(U82(tt, M, N)) → U83¹(isNat(N), M, N)
ACTIVE(U82(tt, M, N)) → ISNAT(N)
ACTIVE(U83(tt, M, N)) → MARK(U84(isNatKind(N), M, N))
ACTIVE(U83(tt, M, N)) → U84¹(isNatKind(N), M, N)
ACTIVE(U83(tt, M, N)) → ISNATKIND(N)
ACTIVE(U84(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(U84(tt, M, N)) → S(plus(N, M))
ACTIVE(U84(tt, M, N)) → PLUS(N, M)
ACTIVE(U91(tt, N)) → MARK(U92(isNatKind(N)))
ACTIVE(U91(tt, N)) → U92¹(isNatKind(N))
ACTIVE(U91(tt, N)) → ISNATKIND(N)
ACTIVE(U92(tt)) → MARK(0)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(plus(V1, V2))) → U11¹(isNatKind(V1), V1, V2)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(isNat(x(V1, V2))) → MARK(U31(isNatKind(V1), V1, V2))
ACTIVE(isNat(x(V1, V2))) → U31¹(isNatKind(V1), V1, V2)
ACTIVE(isNat(x(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(0)) → MARK(tt)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U41(isNatKind(V1), V2))
ACTIVE(isNatKind(plus(V1, V2))) → U41¹(isNatKind(V1), V2)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(s(V1))) → MARK(U51(isNatKind(V1)))
ACTIVE(isNatKind(s(V1))) → U51¹(isNatKind(V1))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatKind(x(V1, V2))) → MARK(U61(isNatKind(V1), V2))
ACTIVE(isNatKind(x(V1, V2))) → U61¹(isNatKind(V1), V2)
ACTIVE(isNatKind(x(V1, V2))) → ISNATKIND(V1)
ACTIVE(plus(N, 0)) → MARK(U71(isNat(N), N))
ACTIVE(plus(N, 0)) → U71¹(isNat(N), N)
ACTIVE(plus(N, 0)) → ISNAT(N)
ACTIVE(plus(N, s(M))) → MARK(U81(isNat(M), M, N))
ACTIVE(plus(N, s(M))) → U81¹(isNat(M), M, N)
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(x(N, 0)) → MARK(U91(isNat(N), N))
ACTIVE(x(N, 0)) → U91¹(isNat(N), N)
ACTIVE(x(N, 0)) → ISNAT(N)
ACTIVE(x(N, s(M))) → MARK(U101(isNat(M), M, N))
ACTIVE(x(N, s(M))) → U101¹(isNat(M), M, N)
ACTIVE(x(N, s(M))) → ISNAT(M)
MARK(U101(X1, X2, X3)) → ACTIVE(U101(mark(X1), X2, X3))
MARK(U101(X1, X2, X3)) → U101¹(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U102(X1, X2, X3)) → ACTIVE(U102(mark(X1), X2, X3))
MARK(U102(X1, X2, X3)) → U102¹(mark(X1), X2, X3)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U103(X1, X2, X3)) → ACTIVE(U103(mark(X1), X2, X3))
MARK(U103(X1, X2, X3)) → U103¹(mark(X1), X2, X3)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U104(X1, X2, X3)) → ACTIVE(U104(mark(X1), X2, X3))
MARK(U104(X1, X2, X3)) → U104¹(mark(X1), X2, X3)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
MARK(x(X1, X2)) → X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → U12¹(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → U13¹(mark(X1), X2, X3)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → U14¹(mark(X1), X2, X3)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U15(X1, X2)) → U15¹(mark(X1), X2)
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → ACTIVE(U16(mark(X)))
MARK(U16(X)) → U16¹(mark(X))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → U23¹(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
MARK(U31(X1, X2, X3)) → U31¹(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2, X3)) → ACTIVE(U32(mark(X1), X2, X3))
MARK(U32(X1, X2, X3)) → U32¹(mark(X1), X2, X3)
MARK(U32(X1, X2, X3)) → MARK(X1)
MARK(U33(X1, X2, X3)) → ACTIVE(U33(mark(X1), X2, X3))
MARK(U33(X1, X2, X3)) → U33¹(mark(X1), X2, X3)
MARK(U33(X1, X2, X3)) → MARK(X1)
MARK(U34(X1, X2, X3)) → ACTIVE(U34(mark(X1), X2, X3))
MARK(U34(X1, X2, X3)) → U34¹(mark(X1), X2, X3)
MARK(U34(X1, X2, X3)) → MARK(X1)
MARK(U35(X1, X2)) → ACTIVE(U35(mark(X1), X2))
MARK(U35(X1, X2)) → U35¹(mark(X1), X2)
MARK(U35(X1, X2)) → MARK(X1)
MARK(U36(X)) → ACTIVE(U36(mark(X)))
MARK(U36(X)) → U36¹(mark(X))
MARK(U36(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U41(X1, X2)) → U41¹(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U42(X)) → U42¹(mark(X))
MARK(U42(X)) → MARK(X)
MARK(U51(X)) → ACTIVE(U51(mark(X)))
MARK(U51(X)) → U51¹(mark(X))
MARK(U51(X)) → MARK(X)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U61(X1, X2)) → U61¹(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U62(X)) → U62¹(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U71(X1, X2)) → U71¹(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U72(X1, X2)) → U72¹(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
MARK(U81(X1, X2, X3)) → U81¹(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
MARK(U82(X1, X2, X3)) → U82¹(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → ACTIVE(U83(mark(X1), X2, X3))
MARK(U83(X1, X2, X3)) → U83¹(mark(X1), X2, X3)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → ACTIVE(U84(mark(X1), X2, X3))
MARK(U84(X1, X2, X3)) → U84¹(mark(X1), X2, X3)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2)) → ACTIVE(U91(mark(X1), X2))
MARK(U91(X1, X2)) → U91¹(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → ACTIVE(U92(mark(X)))
MARK(U92(X)) → U92¹(mark(X))
MARK(U92(X)) → MARK(X)
MARK(0) → ACTIVE(0)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)
ISNATKIND(mark(X)) → ISNATKIND(X)
ISNATKIND(active(X)) → ISNATKIND(X)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)
U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, mark(X2), X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, active(X2), X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)
U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, mark(X2), X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, active(X2), X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)
U15¹(mark(X1), X2) → U15¹(X1, X2)
U15¹(X1, mark(X2)) → U15¹(X1, X2)
U15¹(active(X1), X2) → U15¹(X1, X2)
U15¹(X1, active(X2)) → U15¹(X1, X2)
U16¹(mark(X)) → U16¹(X)
U16¹(active(X)) → U16¹(X)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U23¹(mark(X)) → U23¹(X)
U23¹(active(X)) → U23¹(X)
U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)
U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, mark(X2), X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, mark(X3)) → U32¹(X1, X2, X3)
U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, active(X2), X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, active(X3)) → U32¹(X1, X2, X3)
U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, mark(X2), X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, mark(X3)) → U33¹(X1, X2, X3)
U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, active(X2), X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, active(X3)) → U33¹(X1, X2, X3)
U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, mark(X2), X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, mark(X3)) → U34¹(X1, X2, X3)
U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, active(X2), X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, active(X3)) → U34¹(X1, X2, X3)
U35¹(mark(X1), X2) → U35¹(X1, X2)
U35¹(X1, mark(X2)) → U35¹(X1, X2)
U35¹(active(X1), X2) → U35¹(X1, X2)
U35¹(X1, active(X2)) → U35¹(X1, X2)
U36¹(mark(X)) → U36¹(X)
U36¹(active(X)) → U36¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)
U42¹(mark(X)) → U42¹(X)
U42¹(active(X)) → U42¹(X)
U51¹(mark(X)) → U51¹(X)
U51¹(active(X)) → U51¹(X)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(active(X1), X2) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)
U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(X)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)
U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)
U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)
S(mark(X)) → S(X)
S(active(X)) → S(X)
U91¹(mark(X1), X2) → U91¹(X1, X2)
U91¹(X1, mark(X2)) → U91¹(X1, X2)
U91¹(active(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)
U92¹(mark(X)) → U92¹(X)
U92¹(active(X)) → U92¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(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 38 SCCs with 114 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

U92¹(active(X)) → U92¹(X)
U92¹(mark(X)) → U92¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(active(X)) → U92¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1)  =  U92¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U92^1₁

Status:

active₁: [1]
U92^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

U92¹(mark(X)) → U92¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(mark(X)) → U92¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

U91¹(X1, mark(X2)) → U91¹(X1, X2)
U91¹(mark(X1), X2) → U91¹(X1, X2)
U91¹(active(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(active(X1), X2) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U91^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

U91¹(X1, mark(X2)) → U91¹(X1, X2)
U91¹(mark(X1), X2) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(mark(X1), X2) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U91^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

U91¹(X1, mark(X2)) → U91¹(X1, X2)
U91¹(X1, active(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, active(X2)) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U91^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

U91¹(X1, mark(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, mark(X2)) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U91^1₂

Status:

U91^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(21) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(22) TRUE

(23) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > S₁

Status:

active₁: [1]
S₁: [1]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) Obligation:

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

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U84^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U84^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

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

U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

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

U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

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

U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U84^1₂

Status:

mark₁: [1]
U84^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(40) Obligation:

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

U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(43) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(44) TRUE

(45) Obligation:

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

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U83^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(47) Obligation:

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

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U83^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(49) Obligation:

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

U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(51) Obligation:

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

U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U83^1₂

Status:

U83^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(58) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(59) TRUE

(60) Obligation:

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

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U82^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
mark₁: [1]
U82^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

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

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U82^1₂

Status:

mark₁: [1]
U82^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

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

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(73) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(74) TRUE

(75) Obligation:

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

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U81^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U81^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(79) Obligation:

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

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

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

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U81^1₂

Status:

U81^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

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

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(88) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(89) TRUE

(90) Obligation:

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

U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(91) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(active(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U72^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(92) Obligation:

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

U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
mark₁: [1]
U72^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

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

U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(X1, active(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U72^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

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

U72¹(X1, mark(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(X1, mark(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U72^1₂

Status:

mark₁: [1]
U72^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(98) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(99) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(100) TRUE

(101) Obligation:

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

U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U71^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

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

U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U71^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

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

U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, active(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U71^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

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

U71¹(X1, mark(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, mark(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U71^1₂

Status:

U71^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(110) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(111) TRUE

(112) Obligation:

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

U62¹(active(X)) → U62¹(X)
U62¹(mark(X)) → U62¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(active(X)) → U62¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1)  =  U62¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U62^1₁

Status:

active₁: [1]
U62^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

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

U62¹(mark(X)) → U62¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X)) → U62¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(117) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(118) TRUE

(119) Obligation:

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

U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(active(X1), X2) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X1), X2) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U61^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(121) Obligation:

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

U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X1), X2) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U61^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

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

U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, active(X2)) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U61^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(125) Obligation:

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

U61¹(X1, mark(X2)) → U61¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2)) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U61^1₂

Status:

U61^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(127) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(128) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(129) TRUE

(130) Obligation:

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

U51¹(active(X)) → U51¹(X)
U51¹(mark(X)) → U51¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(active(X)) → U51¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1)  =  U51¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U51^1₁

Status:

active₁: [1]
U51^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

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

U51¹(mark(X)) → U51¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(133) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X)) → U51¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(135) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(136) TRUE

(137) Obligation:

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

U42¹(active(X)) → U42¹(X)
U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(138) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(active(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1)  =  U42¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U42^1₁

Status:

active₁: [1]
U42^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(139) Obligation:

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

U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(141) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(142) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(143) TRUE

(144) Obligation:

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

U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(145) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(active(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U41^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(146) Obligation:

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

U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(147) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U41^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(148) Obligation:

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

U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, active(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U41^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

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

U41¹(X1, mark(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(151) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, mark(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U41^1₂

Status:

U41^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(152) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(153) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(154) TRUE

(155) Obligation:

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

U36¹(active(X)) → U36¹(X)
U36¹(mark(X)) → U36¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(156) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U36¹(active(X)) → U36¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U36¹(x1)  =  U36¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U36^1₁

Status:

active₁: [1]
U36^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(157) Obligation:

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

U36¹(mark(X)) → U36¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U36¹(mark(X)) → U36¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U36¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(159) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(160) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(161) TRUE

(162) Obligation:

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

U35¹(X1, mark(X2)) → U35¹(X1, X2)
U35¹(mark(X1), X2) → U35¹(X1, X2)
U35¹(active(X1), X2) → U35¹(X1, X2)
U35¹(X1, active(X2)) → U35¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(163) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U35¹(active(X1), X2) → U35¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U35¹(x1, x2)  =  U35¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U35^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(164) Obligation:

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

U35¹(X1, mark(X2)) → U35¹(X1, X2)
U35¹(mark(X1), X2) → U35¹(X1, X2)
U35¹(X1, active(X2)) → U35¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(165) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U35¹(mark(X1), X2) → U35¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U35¹(x1, x2)  =  U35¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U35^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(166) Obligation:

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

U35¹(X1, mark(X2)) → U35¹(X1, X2)
U35¹(X1, active(X2)) → U35¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(167) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U35¹(X1, active(X2)) → U35¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U35¹(x1, x2)  =  U35¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U35^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(168) Obligation:

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

U35¹(X1, mark(X2)) → U35¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(169) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U35¹(X1, mark(X2)) → U35¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U35^1₂

Status:

U35^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(170) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(171) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(172) TRUE

(173) Obligation:

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

U34¹(X1, mark(X2), X3) → U34¹(X1, X2, X3)
U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, mark(X3)) → U34¹(X1, X2, X3)
U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, active(X2), X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, active(X3)) → U34¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(174) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U34¹(X1, active(X2), X3) → U34¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  U34¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U34^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(175) Obligation:

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

U34¹(X1, mark(X2), X3) → U34¹(X1, X2, X3)
U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, mark(X3)) → U34¹(X1, X2, X3)
U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, active(X3)) → U34¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(176) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U34¹(X1, mark(X2), X3) → U34¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  U34¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
mark₁: [1]
U34^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(177) Obligation:

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

U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, mark(X3)) → U34¹(X1, X2, X3)
U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, active(X3)) → U34¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(178) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U34¹(X1, X2, mark(X3)) → U34¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(179) Obligation:

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

U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(X1, X2, active(X3)) → U34¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U34¹(X1, X2, active(X3)) → U34¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(181) Obligation:

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

U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)
U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(182) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U34¹(mark(X1), X2, X3) → U34¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  U34¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U34^1₂

Status:

U34^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(183) Obligation:

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

U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U34¹(active(X1), X2, X3) → U34¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(185) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(186) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(187) TRUE

(188) Obligation:

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

U33¹(X1, mark(X2), X3) → U33¹(X1, X2, X3)
U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, mark(X3)) → U33¹(X1, X2, X3)
U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, active(X2), X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, active(X3)) → U33¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(189) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(X1, active(X2), X3) → U33¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  U33¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U33^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(190) Obligation:

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

U33¹(X1, mark(X2), X3) → U33¹(X1, X2, X3)
U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, mark(X3)) → U33¹(X1, X2, X3)
U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, active(X3)) → U33¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(191) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(X1, mark(X2), X3) → U33¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  U33¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U33^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(192) Obligation:

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

U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, mark(X3)) → U33¹(X1, X2, X3)
U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, active(X3)) → U33¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(193) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(X1, X2, mark(X3)) → U33¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(194) Obligation:

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

U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(X1, X2, active(X3)) → U33¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(195) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(X1, X2, active(X3)) → U33¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(196) Obligation:

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

U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)
U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(197) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(mark(X1), X2, X3) → U33¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  U33¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U33^1₂

Status:

mark₁: [1]
U33^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(198) Obligation:

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

U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(199) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(active(X1), X2, X3) → U33¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(200) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(201) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(202) TRUE

(203) Obligation:

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

U32¹(X1, mark(X2), X3) → U32¹(X1, X2, X3)
U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, mark(X3)) → U32¹(X1, X2, X3)
U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, active(X2), X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, active(X3)) → U32¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(204) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, active(X2), X3) → U32¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  U32¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U32^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(205) Obligation:

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

U32¹(X1, mark(X2), X3) → U32¹(X1, X2, X3)
U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, mark(X3)) → U32¹(X1, X2, X3)
U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, active(X3)) → U32¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(206) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, mark(X2), X3) → U32¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  U32¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U32^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(207) Obligation:

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

U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, mark(X3)) → U32¹(X1, X2, X3)
U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, active(X3)) → U32¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(208) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, X2, mark(X3)) → U32¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(209) Obligation:

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

U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(X1, X2, active(X3)) → U32¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(210) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, X2, active(X3)) → U32¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)
U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(212) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(mark(X1), X2, X3) → U32¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  U32¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U32^1₂

Status:

U32^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(213) Obligation:

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

U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(214) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(active(X1), X2, X3) → U32¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(215) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(216) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(217) TRUE

(218) Obligation:

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

U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(219) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, active(X2), X3) → U31¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  U31¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U31^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(220) Obligation:

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

U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)
U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2), X3) → U31¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  U31¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U31^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(222) Obligation:

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

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(223) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, X2, mark(X3)) → U31¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(224) Obligation:

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

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(225) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, X2, active(X3)) → U31¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(226) Obligation:

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

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)
U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(227) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2, X3) → U31¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  U31¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U31^1₂

Status:

U31^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(228) Obligation:

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

U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(229) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X1), X2, X3) → U31¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(230) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(231) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(232) TRUE

(233) Obligation:

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

U23¹(active(X)) → U23¹(X)
U23¹(mark(X)) → U23¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(234) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(active(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  U23¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U23^1₁

Status:

active₁: [1]
U23^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(235) Obligation:

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

U23¹(mark(X)) → U23¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(236) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(mark(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(237) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(238) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(239) TRUE

(240) Obligation:

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

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(241) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(active(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U22^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(242) Obligation:

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

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(243) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U22^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(244) Obligation:

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

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(245) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, active(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U22^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(246) Obligation:

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

U22¹(X1, mark(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(247) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, mark(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U22^1₂

Status:

U22^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(248) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(249) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(250) TRUE

(251) Obligation:

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

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(252) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U21^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(253) Obligation:

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

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(254) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U21^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(255) Obligation:

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

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(256) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, active(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U21^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(257) Obligation:

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

U21¹(X1, mark(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(258) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U21^1₂

Status:

U21^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(259) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(260) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(261) TRUE

(262) Obligation:

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

U16¹(active(X)) → U16¹(X)
U16¹(mark(X)) → U16¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(263) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U16¹(active(X)) → U16¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U16¹(x1)  =  U16¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > U16^1₁

Status:

active₁: [1]
U16^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(264) Obligation:

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

U16¹(mark(X)) → U16¹(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(265) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U16¹(mark(X)) → U16¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U16¹(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(266) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(267) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(268) TRUE

(269) Obligation:

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

U15¹(X1, mark(X2)) → U15¹(X1, X2)
U15¹(mark(X1), X2) → U15¹(X1, X2)
U15¹(active(X1), X2) → U15¹(X1, X2)
U15¹(X1, active(X2)) → U15¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(270) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U15¹(active(X1), X2) → U15¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U15¹(x1, x2)  =  U15¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U15^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(271) Obligation:

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

U15¹(X1, mark(X2)) → U15¹(X1, X2)
U15¹(mark(X1), X2) → U15¹(X1, X2)
U15¹(X1, active(X2)) → U15¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(272) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U15¹(mark(X1), X2) → U15¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U15¹(x1, x2)  =  U15¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U15^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(273) Obligation:

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

U15¹(X1, mark(X2)) → U15¹(X1, X2)
U15¹(X1, active(X2)) → U15¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(274) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U15¹(X1, active(X2)) → U15¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U15¹(x1, x2)  =  U15¹(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U15^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(275) Obligation:

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

U15¹(X1, mark(X2)) → U15¹(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U15¹(X1, mark(X2)) → U15¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > U15^1₂

Status:

U15^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(277) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(278) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(279) TRUE

(280) Obligation:

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

U14¹(X1, mark(X2), X3) → U14¹(X1, X2, X3)
U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, active(X2), X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(281) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14¹(X1, active(X2), X3) → U14¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  U14¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U14^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(282) Obligation:

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

U14¹(X1, mark(X2), X3) → U14¹(X1, X2, X3)
U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(283) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14¹(X1, mark(X2), X3) → U14¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  U14¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
mark₁: [1]
U14^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(284) Obligation:

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

U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(285) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(286) Obligation:

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

U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(287) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(288) Obligation:

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

U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(289) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  U14¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U14^1₂

Status:

mark₁: [1]
U14^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(290) Obligation:

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

U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(291) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(292) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(293) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(294) TRUE

(295) Obligation:

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

U13¹(X1, mark(X2), X3) → U13¹(X1, X2, X3)
U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, active(X2), X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(296) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(X1, active(X2), X3) → U13¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  U13¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U13^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(297) Obligation:

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

U13¹(X1, mark(X2), X3) → U13¹(X1, X2, X3)
U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(298) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(X1, mark(X2), X3) → U13¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  U13¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
mark₁: [1]
U13^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(299) Obligation:

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

U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(300) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(301) Obligation:

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

U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(302) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(303) Obligation:

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

U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(304) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  U13¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U13^1₂

Status:

mark₁: [1]
U13^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(305) Obligation:

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

U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(306) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(307) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(308) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(309) TRUE

(310) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(311) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U12^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(312) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(313) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U12^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(314) Obligation:

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

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(315) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(316) Obligation:

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

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(317) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(318) Obligation:

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

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(319) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U12^1₂

Status:

U12^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(320) Obligation:

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

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(321) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(322) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(323) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(324) TRUE

(325) Obligation:

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

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(326) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U11^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(327) Obligation:

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

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(328) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U11^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(329) Obligation:

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(330) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(331) Obligation:

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(332) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(333) Obligation:

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(334) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U11^1₂

Status:

U11^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(335) Obligation:

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

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(336) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(337) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(338) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(339) TRUE

(340) Obligation:

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

X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(active(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(341) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(active(X1), X2) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

X₁: [1]
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(342) Obligation:

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

X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(343) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(mark(X1), X2) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

X₁: [1]
active₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(344) Obligation:

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

X(X1, mark(X2)) → X(X1, X2)
X(X1, active(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(345) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(X1, active(X2)) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
X₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(346) Obligation:

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

X(X1, mark(X2)) → X(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(347) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(X1, mark(X2)) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > X₂

Status:

X₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(348) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(349) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(350) TRUE

(351) Obligation:

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

PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(352) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(active(X1), X2) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
PLUS₁: [1]

The following usable rules [FROCOS05] were oriented: none

(353) Obligation:

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

PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(354) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(mark(X1), X2) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
mark₁: [1]
PLUS₁: [1]

The following usable rules [FROCOS05] were oriented: none

(355) Obligation:

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

PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(356) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, active(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1, x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
PLUS₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(357) Obligation:

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

PLUS(X1, mark(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(358) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, mark(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:

mark₁ > PLUS₂

Status:

PLUS₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(359) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(360) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(361) TRUE

(362) Obligation:

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

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(363) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  U104¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U104^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(364) Obligation:

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

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(365) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  U104¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U104^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(366) Obligation:

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

U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(367) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(368) Obligation:

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

U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(369) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(370) Obligation:

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

U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(371) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  U104¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U104^1₂

Status:

U104^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(372) Obligation:

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

U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(373) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(374) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(375) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(376) TRUE

(377) Obligation:

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(378) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(active(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > ISNAT₁

Status:

active₁: [1]
ISNAT₁: [1]

The following usable rules [FROCOS05] were oriented: none

(379) Obligation:

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

ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(380) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(mark(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(381) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(382) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(383) TRUE

(384) Obligation:

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

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(385) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  U103¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U103^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(386) Obligation:

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

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(387) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  U103¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U103^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(388) Obligation:

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

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(389) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(390) Obligation:

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

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(391) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(392) Obligation:

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

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(393) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  U103¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U103^1₂

Status:

mark₁: [1]
U103^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(394) Obligation:

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

U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(395) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(396) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(397) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(398) TRUE

(399) Obligation:

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

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(400) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(active(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

active₁ > ISNATKIND₁

Status:

active₁: [1]
ISNATKIND₁: [1]

The following usable rules [FROCOS05] were oriented: none

(401) Obligation:

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

ISNATKIND(mark(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(402) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(mark(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(403) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(404) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(405) TRUE

(406) Obligation:

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

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(407) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  U102¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U102^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(408) Obligation:

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

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(409) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  U102¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U102^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(410) Obligation:

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

U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(411) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(412) Obligation:

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

U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(413) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(414) Obligation:

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

U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(415) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)

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)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U102^1₂

Status:

U102^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(416) Obligation:

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

U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(417) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(418) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(419) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(420) TRUE

(421) Obligation:

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

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(422) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U101^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(423) Obligation:

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

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(424) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active: []
U101^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(425) Obligation:

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

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(426) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(427) Obligation:

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

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(428) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(429) Obligation:

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

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(430) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U101^1₂

Status:

mark₁: [1]
U101^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(431) Obligation:

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

U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(432) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(433) Obligation:

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

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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

(434) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(435) TRUE

(436) Obligation:

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

MARK(U101(X1, X2, X3)) → ACTIVE(U101(mark(X1), X2, X3))
ACTIVE(U101(tt, M, N)) → MARK(U102(isNatKind(M), M, N))
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → ACTIVE(U102(mark(X1), X2, X3))
ACTIVE(U102(tt, M, N)) → MARK(U103(isNat(N), M, N))
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U103(tt, M, N)) → MARK(U104(isNatKind(N), M, N))
MARK(U103(X1, X2, X3)) → ACTIVE(U103(mark(X1), X2, X3))
ACTIVE(U104(tt, M, N)) → MARK(plus(x(N, M), N))
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U104(X1, X2, X3)) → ACTIVE(U104(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(x(X1, X2)) → ACTIVE(x(mark(X1), mark(X2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U31(tt, V1, V2)) → MARK(U32(isNatKind(V1), V1, V2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U32(tt, V1, V2)) → MARK(U33(isNatKind(V2), V1, V2))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U33(tt, V1, V2)) → MARK(U34(isNatKind(V2), V1, V2))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U34(tt, V1, V2)) → MARK(U35(isNat(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U35(tt, V2)) → MARK(U36(isNat(V2)))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatKind(V2)))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2, X3)) → ACTIVE(U32(mark(X1), X2, X3))
ACTIVE(U71(tt, N)) → MARK(U72(isNatKind(N), N))
MARK(U32(X1, X2, X3)) → MARK(X1)
MARK(U33(X1, X2, X3)) → ACTIVE(U33(mark(X1), X2, X3))
ACTIVE(U72(tt, N)) → MARK(N)
MARK(U33(X1, X2, X3)) → MARK(X1)
MARK(U34(X1, X2, X3)) → ACTIVE(U34(mark(X1), X2, X3))
ACTIVE(U81(tt, M, N)) → MARK(U82(isNatKind(M), M, N))
MARK(U34(X1, X2, X3)) → MARK(X1)
MARK(U35(X1, X2)) → ACTIVE(U35(mark(X1), X2))
ACTIVE(U82(tt, M, N)) → MARK(U83(isNat(N), M, N))
MARK(U35(X1, X2)) → MARK(X1)
MARK(U36(X)) → ACTIVE(U36(mark(X)))
ACTIVE(U83(tt, M, N)) → MARK(U84(isNatKind(N), M, N))
MARK(U36(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U84(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
ACTIVE(U91(tt, N)) → MARK(U92(isNatKind(N)))
MARK(U42(X)) → MARK(X)
MARK(U51(X)) → ACTIVE(U51(mark(X)))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U51(X)) → MARK(X)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
ACTIVE(isNat(x(V1, V2))) → MARK(U31(isNatKind(V1), V1, V2))
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U41(isNatKind(V1), V2))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(isNatKind(s(V1))) → MARK(U51(isNatKind(V1)))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
ACTIVE(isNatKind(x(V1, V2))) → MARK(U61(isNatKind(V1), V2))
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
ACTIVE(plus(N, 0)) → MARK(U71(isNat(N), N))
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → ACTIVE(U83(mark(X1), X2, X3))
ACTIVE(plus(N, s(M))) → MARK(U81(isNat(M), M, N))
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → ACTIVE(U84(mark(X1), X2, X3))
ACTIVE(x(N, 0)) → MARK(U91(isNat(N), N))
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(x(N, s(M))) → MARK(U101(isNat(M), M, N))
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2)) → ACTIVE(U91(mark(X1), X2))
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → ACTIVE(U92(mark(X)))
MARK(U92(X)) → MARK(X)

The TRS R consists of the following rules:

active(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNat(X)) → active(isNat(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
mark(U16(X)) → active(U16(mark(X)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2, X3)) → active(U32(mark(X1), X2, X3))
mark(U33(X1, X2, X3)) → active(U33(mark(X1), X2, X3))
mark(U34(X1, X2, X3)) → active(U34(mark(X1), X2, X3))
mark(U35(X1, X2)) → active(U35(mark(X1), X2))
mark(U36(X)) → active(U36(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X)) → active(U51(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(U91(X1, X2)) → active(U91(mark(X1), X2))
mark(U92(X)) → active(U92(mark(X)))
mark(0) → active(0)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
x(mark(X1), X2) → x(X1, X2)
x(X1, mark(X2)) → x(X1, X2)
x(active(X1), X2) → x(X1, X2)
x(X1, active(X2)) → x(X1, X2)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U14(mark(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, active(X2), X3) → U14(X1, X2, X3)
U14(X1, X2, active(X3)) → U14(X1, X2, X3)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, mark(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, mark(X3)) → U32(X1, X2, X3)
U32(active(X1), X2, X3) → U32(X1, X2, X3)
U32(X1, active(X2), X3) → U32(X1, X2, X3)
U32(X1, X2, active(X3)) → U32(X1, X2, X3)
U33(mark(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, mark(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, mark(X3)) → U33(X1, X2, X3)
U33(active(X1), X2, X3) → U33(X1, X2, X3)
U33(X1, active(X2), X3) → U33(X1, X2, X3)
U33(X1, X2, active(X3)) → U33(X1, X2, X3)
U34(mark(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, mark(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, mark(X3)) → U34(X1, X2, X3)
U34(active(X1), X2, X3) → U34(X1, X2, X3)
U34(X1, active(X2), X3) → U34(X1, X2, X3)
U34(X1, X2, active(X3)) → U34(X1, X2, X3)
U35(mark(X1), X2) → U35(X1, X2)
U35(X1, mark(X2)) → U35(X1, X2)
U35(active(X1), X2) → U35(X1, X2)
U35(X1, active(X2)) → U35(X1, X2)
U36(mark(X)) → U36(X)
U36(active(X)) → U36(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X)) → U51(X)
U51(active(X)) → U51(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U91(mark(X1), X2) → U91(X1, X2)
U91(X1, mark(X2)) → U91(X1, X2)
U91(active(X1), X2) → U91(X1, X2)
U91(X1, active(X2)) → U91(X1, X2)
U92(mark(X)) → U92(X)
U92(active(X)) → U92(X)

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