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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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(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, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U31(tt, V2)) → U32¹(isNatKind(V2))
ACTIVE(U31(tt, V2)) → ISNATKIND(V2)
ACTIVE(U32(tt)) → MARK(tt)
ACTIVE(U41(tt)) → MARK(tt)
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
ACTIVE(U51(tt, N)) → U52¹(isNatKind(N), N)
ACTIVE(U51(tt, N)) → ISNATKIND(N)
ACTIVE(U52(tt, N)) → MARK(N)
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
ACTIVE(U61(tt, M, N)) → U62¹(isNatKind(M), M, N)
ACTIVE(U61(tt, M, N)) → ISNATKIND(M)
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U62(tt, M, N)) → U63¹(isNat(N), M, N)
ACTIVE(U62(tt, M, N)) → ISNAT(N)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(U63(tt, M, N)) → U64¹(isNatKind(N), M, N)
ACTIVE(U63(tt, M, N)) → ISNATKIND(N)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
ACTIVE(U64(tt, M, N)) → S(plus(N, M))
ACTIVE(U64(tt, M, N)) → PLUS(N, M)
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(isNatKind(0)) → MARK(tt)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
ACTIVE(isNatKind(plus(V1, V2))) → U31¹(isNatKind(V1), V2)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
ACTIVE(isNatKind(s(V1))) → U41¹(isNatKind(V1))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
ACTIVE(plus(N, 0)) → U51¹(isNat(N), N)
ACTIVE(plus(N, 0)) → ISNAT(N)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(plus(N, s(M))) → U61¹(isNat(M), M, N)
ACTIVE(plus(N, s(M))) → ISNAT(M)
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(tt) → ACTIVE(tt)
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(isNatKind(X)) → ACTIVE(isNatKind(X))
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(isNat(X)) → ACTIVE(isNat(X))
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)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U32(X)) → U32¹(mark(X))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → U41¹(mark(X))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U51(X1, X2)) → U51¹(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U52(X1, X2)) → U52¹(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U61(X1, X2, X3)) → U61¹(mark(X1), X2, X3)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(U62(X1, X2, X3)) → U62¹(mark(X1), X2, X3)
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U63(X1, X2, X3)) → U63¹(mark(X1), X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → U64¹(mark(X1), X2, X3)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
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(0) → ACTIVE(0)
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)
ISNATKIND(mark(X)) → ISNATKIND(X)
ISNATKIND(active(X)) → ISNATKIND(X)
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)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
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) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)
U32¹(mark(X)) → U32¹(X)
U32¹(active(X)) → U32¹(X)
U41¹(mark(X)) → U41¹(X)
U41¹(active(X)) → U41¹(X)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)
U62¹(mark(X1), X2, X3) → U62¹(X1, X2, X3)
U62¹(X1, mark(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, mark(X3)) → U62¹(X1, X2, X3)
U62¹(active(X1), X2, X3) → U62¹(X1, X2, X3)
U62¹(X1, active(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, active(X3)) → U62¹(X1, X2, X3)
U63¹(mark(X1), X2, X3) → U63¹(X1, X2, X3)
U63¹(X1, mark(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, mark(X3)) → U63¹(X1, X2, X3)
U63¹(active(X1), X2, X3) → U63¹(X1, X2, X3)
U63¹(X1, active(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, active(X3)) → U63¹(X1, X2, X3)
U64¹(mark(X1), X2, X3) → U64¹(X1, X2, X3)
U64¹(X1, mark(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, mark(X3)) → U64¹(X1, X2, X3)
U64¹(active(X1), X2, X3) → U64¹(X1, X2, X3)
U64¹(X1, active(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, active(X3)) → U64¹(X1, X2, X3)
S(mark(X)) → S(X)
S(active(X)) → S(X)
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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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 23 SCCs with 66 less nodes.

(4) Complex Obligation (AND)

(5) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

PLUS₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(7) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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)

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

trivial

Status:

PLUS₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(9) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(10) 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: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x2)
mark(x1)  =  mark(x1)

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

mark₁ > PLUS₁

Status:

PLUS₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

PLUS(mark(X1), X2) → PLUS(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

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

[PLUS₂, mark₁]

Status:

PLUS₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(18) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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)

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

trivial

Status:

mark₁: multiset

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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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:

U64¹(X1, mark(X2), X3) → U64¹(X1, X2, X3)
U64¹(mark(X1), X2, X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, mark(X3)) → U64¹(X1, X2, X3)
U64¹(active(X1), X2, X3) → U64¹(X1, X2, X3)
U64¹(X1, active(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, active(X3)) → U64¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U64¹(mark(X1), X2, X3) → U64¹(X1, X2, X3)
U64¹(active(X1), X2, X3) → U64¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

U64¹(X1, mark(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, mark(X3)) → U64¹(X1, X2, X3)
U64¹(X1, active(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, active(X3)) → U64¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U64¹(X1, mark(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, mark(X3)) → U64¹(X1, X2, X3)
U64¹(X1, active(X2), X3) → U64¹(X1, X2, X3)
U64¹(X1, X2, active(X3)) → U64¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2, x3)  =  U64¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U64^1₂
active₁ > U64^1₂

Status:

U64^1₂: multiset
mark₁: multiset
active₁: multiset

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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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:

U63¹(X1, mark(X2), X3) → U63¹(X1, X2, X3)
U63¹(mark(X1), X2, X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, mark(X3)) → U63¹(X1, X2, X3)
U63¹(active(X1), X2, X3) → U63¹(X1, X2, X3)
U63¹(X1, active(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, active(X3)) → U63¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U63¹(mark(X1), X2, X3) → U63¹(X1, X2, X3)
U63¹(active(X1), X2, X3) → U63¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

U63¹(X1, mark(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, mark(X3)) → U63¹(X1, X2, X3)
U63¹(X1, active(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, active(X3)) → U63¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U63¹(X1, mark(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, mark(X3)) → U63¹(X1, X2, X3)
U63¹(X1, active(X2), X3) → U63¹(X1, X2, X3)
U63¹(X1, X2, active(X3)) → U63¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3)  =  U63¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U63^1₂
active₁ > U63^1₂

Status:

U63^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

U62¹(X1, mark(X2), X3) → U62¹(X1, X2, X3)
U62¹(mark(X1), X2, X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, mark(X3)) → U62¹(X1, X2, X3)
U62¹(active(X1), X2, X3) → U62¹(X1, X2, X3)
U62¹(X1, active(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, active(X3)) → U62¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X1), X2, X3) → U62¹(X1, X2, X3)
U62¹(active(X1), X2, X3) → U62¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

U62¹(X1, mark(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, mark(X3)) → U62¹(X1, X2, X3)
U62¹(X1, active(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, active(X3)) → U62¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, mark(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, mark(X3)) → U62¹(X1, X2, X3)
U62¹(X1, active(X2), X3) → U62¹(X1, X2, X3)
U62¹(X1, X2, active(X3)) → U62¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3)  =  U62¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U62^1₂
active₁ > U62^1₂

Status:

U62^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3)  =  U61¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U61^1₂
active₁ > U61^1₂

Status:

U61^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U52¹(X1, active(X2)) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  U52¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U52^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U52¹(active(X1), X2) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  U52¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U52^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U52¹(X1, mark(X2)) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  U52¹(x2)
mark(x1)  =  mark(x1)

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

mark₁ > U52^1₁

Status:

U52^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

U52¹(mark(X1), X2) → U52¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X1), X2) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U52^1₂, mark₁]

Status:

U52^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U51¹(X1, active(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U51^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

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

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U51¹(active(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U51^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U51¹(X1, mark(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x2)
mark(x1)  =  mark(x1)

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

mark₁ > U51^1₁

Status:

U51^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

U51¹(mark(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U51¹(mark(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U51^1₂, mark₁]

Status:

U51^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) Obligation:

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

U41¹(active(X)) → U41¹(X)
U41¹(mark(X)) → U41¹(X)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(active(X)) → U41¹(X)

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

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

U41¹(mark(X)) → U41¹(X)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U41¹(mark(X)) → U41¹(X)

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

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(77) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(78) PisEmptyProof (EQUIVALENT transformation)

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

(79) TRUE

(80) Obligation:

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

U32¹(active(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(active(X)) → U32¹(X)

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

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

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

U32¹(mark(X)) → U32¹(X)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(mark(X)) → U32¹(X)

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

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(84) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(85) PisEmptyProof (EQUIVALENT transformation)

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

(86) TRUE

(87) Obligation:

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

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

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, active(X2)) → U31¹(X1, X2)

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

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

trivial

Status:

U31^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

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

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X1), X2) → U31¹(X1, X2)

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

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

trivial

Status:

U31^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

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

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

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2)) → U31¹(X1, X2)

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

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

mark₁ > U31^1₁

Status:

U31^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(93) Obligation:

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

U31¹(mark(X1), X2) → U31¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U31^1₂, mark₁]

Status:

U31^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(96) PisEmptyProof (EQUIVALENT transformation)

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

(97) TRUE

(98) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(99) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(100) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(101) 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)

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(102) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(103) PisEmptyProof (EQUIVALENT transformation)

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

(104) TRUE

(105) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U22^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(107) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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)

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

trivial

Status:

U22^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(109) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(110) 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: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x2)
mark(x1)  =  mark(x1)

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

mark₁ > U22^1₁

Status:

U22^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(111) Obligation:

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

U22¹(mark(X1), X2) → U22¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

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

[U22^1₂, mark₁]

Status:

U22^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(113) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(114) PisEmptyProof (EQUIVALENT transformation)

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

(115) TRUE

(116) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(117) 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(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U21^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(118) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(119) 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)

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

trivial

Status:

U21^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(120) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(121) 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: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x2)
mark(x1)  =  mark(x1)

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

mark₁ > U21^1₁

Status:

U21^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

U21¹(mark(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

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

[U21^1₂, mark₁]

Status:

U21^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(128) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(129) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(130) 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)

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(131) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(132) PisEmptyProof (EQUIVALENT transformation)

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

(133) TRUE

(134) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(135) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(136) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(137) 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)

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

trivial

Status:

mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(138) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(139) PisEmptyProof (EQUIVALENT transformation)

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

(140) TRUE

(141) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(142) 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(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

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

trivial

Status:

U15^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(143) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(144) 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)

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

trivial

Status:

U15^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(145) 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)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(146) 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: Combined order from the following AFS and order.
U15¹(x1, x2)  =  U15¹(x2)
mark(x1)  =  mark(x1)

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

mark₁ > U15^1₁

Status:

U15^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(147) Obligation:

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

U15¹(mark(X1), X2) → U15¹(X1, X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

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

[U15^1₂, mark₁]

Status:

U15^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(149) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(150) PisEmptyProof (EQUIVALENT transformation)

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

(151) TRUE

(152) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(153) 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)
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
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(154) Obligation:

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

U14¹(X1, mark(X2), X3) → U14¹(X1, X2, X3)
U14¹(X1, X2, mark(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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(155) 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)
U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)
U14¹(X1, active(X2), X3) → U14¹(X1, X2, X3)
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)  =  U14¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U14^1₂
active₁ > U14^1₂

Status:

U14^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(156) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(157) PisEmptyProof (EQUIVALENT transformation)

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

(158) TRUE

(159) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(160) 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)
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
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(161) Obligation:

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

U13¹(X1, mark(X2), X3) → U13¹(X1, X2, X3)
U13¹(X1, X2, mark(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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(162) 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)
U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)
U13¹(X1, active(X2), X3) → U13¹(X1, X2, X3)
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)  =  U13¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U13^1₂
active₁ > U13^1₂

Status:

U13^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(163) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(164) PisEmptyProof (EQUIVALENT transformation)

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

(165) TRUE

(166) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

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

trivial

Status:

active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(168) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

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)

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

trivial

Status:

mark₁: multiset

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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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:

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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
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
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(175) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
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)  =  U12¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U12^1₂
active₁ > U12^1₂

Status:

U12^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(177) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(178) PisEmptyProof (EQUIVALENT transformation)

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

(179) TRUE

(180) 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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(181) 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)
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
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

trivial

Status:

mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(182) Obligation:

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

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(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(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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(183) 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)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
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)  =  U11¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

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

mark₁ > U11^1₂
active₁ > U11^1₂

Status:

U11^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(184) Obligation:

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

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(185) PisEmptyProof (EQUIVALENT transformation)

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

(186) TRUE

(187) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13
U14(x1, x2, x3)  =  U14
U15(x1, x2)  =  U15
isNat(x1)  =  isNat
U16(x1)  =  U16
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
U23(x1)  =  U23
U31(x1, x2)  =  U31
U32(x1)  =  U32
U51(x1, x2)  =  U51
U52(x1, x2)  =  U52
U61(x1, x2, x3)  =  U61
U62(x1, x2, x3)  =  U62
U63(x1, x2, x3)  =  U63
U64(x1, x2, x3)  =  U64
U41(x1)  =  U41
s(x1)  =  s
plus(x1, x2)  =  plus
0  =  0
active(x1)  =  active

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

[tt, 0] > [MARK, U11, mark, U12, isNatKind, U13, U14, U15, isNat, U16, U21, U22, U23, U31, U32, U51, U52, U61, U62, U63, U64, U41, plus, active] > s

Status:

MARK: multiset
U11: multiset
mark: multiset
tt: multiset
U12: multiset
isNatKind: multiset
U13: multiset
U14: multiset
U15: multiset
isNat: multiset
U16: multiset
U21: multiset
U22: multiset
U23: multiset
U31: multiset
U32: multiset
U51: multiset
U52: multiset
U61: multiset
U62: multiset
U63: multiset
U64: multiset
U41: multiset
s: multiset
plus: multiset
0: multiset
active: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)
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)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)

(189) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
ACTIVE(U52(tt, N)) → MARK(N)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → MARK(X1)
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
MARK(U63(X1, X2, X3)) → MARK(X1)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  x1
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
U41(x1)  =  x1
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
active(x1)  =  x1

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

[U61₃, U62₃, U63₃, U64₃, plus₂] > [tt, isNatKind, isNat, s₁] > U52₂
[U61₃, U62₃, U63₃, U64₃, plus₂] > U51₂ > U52₂

Status:

tt: multiset
isNatKind: multiset
isNat: multiset
U51₂: multiset
U52₂: [1,2]
U61₃: multiset
U62₃: multiset
U63₃: multiset
U64₃: multiset
s₁: multiset
plus₂: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
mark(isNatKind(X)) → active(isNatKind(X))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
mark(isNat(X)) → active(isNat(X))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
mark(U16(X)) → active(U16(mark(X)))
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
active(U52(tt, N)) → mark(N)
mark(U23(X)) → active(U23(mark(X)))
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
mark(U32(X)) → active(U32(mark(X)))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
mark(U41(X)) → active(U41(mark(X)))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
active(plus(N, 0)) → mark(U51(isNat(N), N))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(0) → active(0)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)
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)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
active(U16(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(isNatKind(0)) → mark(tt)

(191) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(192) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  U64
U41(x1)  =  x1
U51(x1, x2)  =  x1
plus(x1, x2)  =  plus(x1, x2)
U52(x1, x2)  =  x1
s(x1)  =  s
active(x1)  =  x1
0  =  0

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

[tt, 0] > plus₂ > [ACTIVE, isNatKind, isNat, U64]
[tt, 0] > s > [ACTIVE, isNatKind, isNat, U64]

Status:

ACTIVE: []
tt: multiset
isNatKind: []
isNat: []
U64: []
plus₂: multiset
s: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(193) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(194) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U61(x1, x2, x3)  =  U61
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  U63
U64(x1, x2, x3)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x2)
plus(x1, x2)  =  plus(x1, x2)
U52(x1, x2)  =  U52
s(x1)  =  s(x1)
active(x1)  =  x1
0  =  0

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

tt > plus₂ > [ACTIVE, isNatKind, isNat, U61, U63, U51₁, U52]
tt > s₁ > [ACTIVE, isNatKind, isNat, U61, U63, U51₁, U52]
0 > [ACTIVE, isNatKind, isNat, U61, U63, U51₁, U52]

Status:

ACTIVE: []
tt: multiset
isNatKind: []
isNat: []
U61: []
U63: []
U51₁: multiset
plus₂: multiset
U52: []
s₁: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(195) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(196) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U61(x1, x2, x3)  =  U61
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  U64
U41(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
U52(x1, x2)  =  U52(x2)
s(x1)  =  s(x1)
active(x1)  =  active(x1)
U51(x1, x2)  =  x1
0  =  0

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

[tt, 0] > plus₂ > [ACTIVE, isNatKind, isNat, U61, U64, U52₁, active₁]
[tt, 0] > s₁ > [ACTIVE, isNatKind, isNat, U61, U64, U52₁, active₁]

Status:

ACTIVE: []
tt: multiset
isNatKind: []
isNat: []
U61: []
U64: []
plus₂: multiset
U52₁: [1]
s₁: multiset
active₁: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(197) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U61(x1, x2, x3)  =  U61
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  U63
U64(x1, x2, x3)  =  U64
U41(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
active(x1)  =  active(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
0  =  0

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

[tt, 0] > plus₂ > [ACTIVE, isNatKind, isNat, U63, U64, active₁, U51₂, U52₂]
[tt, 0] > s₁ > [ACTIVE, isNatKind, isNat, U63, U64, active₁, U51₂, U52₂]
U61 > [ACTIVE, isNatKind, isNat, U63, U64, active₁, U51₂, U52₂]

Status:

ACTIVE: []
tt: multiset
isNatKind: []
isNat: []
U61: []
U63: []
U64: []
plus₂: multiset
s₁: multiset
active₁: multiset
U51₂: multiset
U52₂: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(199) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(200) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13
U14(x1, x2, x3)  =  U14
U15(x1, x2)  =  U15
isNat(x1)  =  isNat
U16(x1)  =  U16
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
U23(x1)  =  U23
U31(x1, x2)  =  U31
U32(x1)  =  U32
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  U62
U63(x1, x2, x3)  =  U63
U64(x1, x2, x3)  =  U64
U41(x1)  =  U41
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
active(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1, x2)  =  U52(x1, x2)
0  =  0

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

[tt, 0] > plus₂ > [MARK, U11, mark, U12, isNatKind, U13, U14, U15, isNat, U16, U21, U22, U23, U31, U32, U62, U63, U64, U41, U52₂]
[tt, 0] > s₁ > [MARK, U11, mark, U12, isNatKind, U13, U14, U15, isNat, U16, U21, U22, U23, U31, U32, U62, U63, U64, U41, U52₂]

Status:

MARK: multiset
U11: multiset
mark: multiset
tt: multiset
U12: multiset
isNatKind: multiset
U13: multiset
U14: multiset
U15: multiset
isNat: multiset
U16: multiset
U21: multiset
U22: multiset
U23: multiset
U31: multiset
U32: multiset
U62: multiset
U63: multiset
U64: multiset
U41: multiset
plus₂: multiset
s₁: multiset
U52₂: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(201) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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

(202) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE(x1)
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U62(x1, x2, x3)  =  U62(x1, x3)
U63(x1, x2, x3)  =  U63(x1)
U64(x1, x2, x3)  =  U64
U41(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s
active(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  U61(x1, x3)
0  =  0

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

[U62₂, U63₁, U64, plus₂, U61₂] > s > [tt, isNatKind, isNat] > [MARK₁, ACTIVE₁]
0 > [tt, isNatKind, isNat] > [MARK₁, ACTIVE₁]

Status:

MARK₁: [1]
ACTIVE₁: [1]
tt: multiset
isNatKind: []
isNat: []
U62₂: [1,2]
U63₁: [1]
U64: []
plus₂: [2,1]
s: []
U61₂: [1,2]
0: multiset

The following usable rules [FROCOS05] were oriented:

mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
mark(isNatKind(X)) → active(isNatKind(X))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
mark(U13(X1, X2, X3)) → active(U13(mark(X1), X2, X3))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
mark(U14(X1, X2, X3)) → active(U14(mark(X1), X2, X3))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
mark(U15(X1, X2)) → active(U15(mark(X1), X2))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
mark(isNat(X)) → active(isNat(X))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
mark(U16(X)) → active(U16(mark(X)))
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
active(U52(tt, N)) → mark(N)
mark(U23(X)) → active(U23(mark(X)))
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
mark(U32(X)) → active(U32(mark(X)))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
mark(U41(X)) → active(U41(mark(X)))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
active(plus(N, 0)) → mark(U51(isNat(N), N))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(0) → active(0)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
active(U16(tt)) → mark(tt)
active(U23(tt)) → mark(tt)
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(isNatKind(0)) → mark(tt)
U52(X1, mark(X2)) → U52(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
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)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)

(203) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
U62(x1, x2, x3)  =  U62
U63(x1, x2, x3)  =  U63
U64(x1, x2, x3)  =  U64
active(x1)  =  active(x1)
U51(x1, x2)  =  x1
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  x1
0  =  0

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

[tt, 0] > U64 > plus₂ > [ACTIVE, mark₁, isNatKind, isNat, U62, U63, active₁, U52₂]
[tt, 0] > U64 > s₁ > [ACTIVE, mark₁, isNatKind, isNat, U62, U63, active₁, U52₂]

Status:

ACTIVE: []
mark₁: [1]
tt: multiset
isNatKind: []
isNat: []
plus₂: multiset
s₁: multiset
U62: []
U63: []
U64: multiset
active₁: [1]
U52₂: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(205) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
U62(x1, x2, x3)  =  U62
U63(x1, x2, x3)  =  U63(x3)
active(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  x1
0  =  0

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

[tt, 0] > plus₂ > [ACTIVE, mark₁, isNatKind, isNat, U62, U63₁, U51₂, U52₂]
[tt, 0] > s₁ > [ACTIVE, mark₁, isNatKind, isNat, U62, U63₁, U51₂, U52₂]

Status:

ACTIVE: multiset
mark₁: multiset
tt: multiset
isNatKind: multiset
isNat: multiset
plus₂: multiset
s₁: multiset
U62: multiset
U63₁: multiset
U51₂: multiset
U52₂: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(207) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U11(x1, x2, x3)  =  x1
ACTIVE(x1)  =  ACTIVE
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
U62(x1, x2, x3)  =  U62(x3)
active(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  x1
0  =  0

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

[tt, 0] > plus₂ > [ACTIVE, mark₁, isNatKind, isNat, U62₁, U51₂, U52₂]
[tt, 0] > s₁ > [ACTIVE, mark₁, isNatKind, isNat, U62₁, U51₂, U52₂]

Status:

ACTIVE: multiset
mark₁: multiset
tt: multiset
isNatKind: multiset
isNat: multiset
plus₂: multiset
s₁: multiset
U62₁: multiset
U51₂: multiset
U52₂: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(209) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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.

MARK(U16(X)) → ACTIVE(U16(mark(X)))
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U41(X)) → ACTIVE(U41(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark
tt  =  tt
U12(x1, x2, x3)  =  U12
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13
U14(x1, x2, x3)  =  U14
U15(x1, x2)  =  U15
isNat(x1)  =  isNat
U16(x1)  =  U16
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
U23(x1)  =  U23
U31(x1, x2)  =  U31
U32(x1)  =  U32
U41(x1)  =  U41
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  x1
active(x1)  =  active
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
0  =  0

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

0 > [mark, active] > tt > [MARK, U11, U12, isNatKind, U13, U14, U15, isNat, U21, U22, U31] > U16 > [U41, U51₂]
0 > [mark, active] > tt > [MARK, U11, U12, isNatKind, U13, U14, U15, isNat, U21, U22, U31] > U23 > [U41, U51₂]
0 > [mark, active] > tt > [MARK, U11, U12, isNatKind, U13, U14, U15, isNat, U21, U22, U31] > U32 > [U41, U51₂]
0 > [mark, active] > tt > [plus₂, U61₃, U62₃, U63₃, U64₃] > [U41, U51₂]

Status:

MARK: multiset
U11: multiset
mark: []
tt: multiset
U12: multiset
isNatKind: multiset
U13: multiset
U14: multiset
U15: multiset
isNat: multiset
U16: multiset
U21: multiset
U22: multiset
U23: multiset
U31: multiset
U32: multiset
U41: multiset
plus₂: multiset
active: []
U51₂: multiset
U61₃: multiset
U62₃: multiset
U63₃: multiset
U64₃: multiset
0: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), 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)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), 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(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(mark(X1), 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(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(mark(X1), 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)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(X1, mark(X2)) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(active(X1), X2) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U16(active(X)) → U16(X)
U16(mark(X)) → U16(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(active(X)) → U23(X)
U23(mark(X)) → U23(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(active(X)) → U41(X)
U41(mark(X)) → U41(X)

(211) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
MARK(U15(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))

The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
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(isNat(X)) → active(isNat(X))
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)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X)) → active(U41(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3)) → active(U61(mark(X1), X2, X3))
mark(U62(X1, X2, X3)) → active(U62(mark(X1), X2, X3))
mark(U63(X1, X2, X3)) → active(U63(mark(X1), X2, X3))
mark(U64(X1, X2, X3)) → active(U64(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
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)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U63(mark(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, mark(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, mark(X3)) → U63(X1, X2, X3)
U63(active(X1), X2, X3) → U63(X1, X2, X3)
U63(X1, active(X2), X3) → U63(X1, X2, X3)
U63(X1, X2, active(X3)) → U63(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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)

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