Termination w.r.t. Q of the following Term Rewriting System could be proven:

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.


QTRS
  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U161(active(X)) → U161(X)
U641(X1, mark(X2), X3) → U641(X1, X2, X3)
U121(X1, mark(X2), X3) → U121(X1, X2, X3)
ACTIVE(U63(tt, M, N)) → ISNATKIND(N)
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
U631(X1, active(X2), X3) → U631(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
ACTIVE(U61(tt, M, N)) → ISNATKIND(M)
MARK(U15(X1, X2)) → U151(mark(X1), X2)
ACTIVE(U22(tt, V1)) → ISNAT(V1)
U151(mark(X1), X2) → U151(X1, X2)
U311(mark(X1), X2) → U311(X1, X2)
U311(X1, mark(X2)) → U311(X1, X2)
MARK(U32(X)) → MARK(X)
MARK(U13(X1, X2, X3)) → U131(mark(X1), X2, X3)
ACTIVE(U12(tt, V1, V2)) → U131(isNatKind(V2), V1, V2)
ACTIVE(U51(tt, N)) → ISNATKIND(N)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
U641(X1, X2, mark(X3)) → U641(X1, X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → MARK(X1)
U611(X1, X2, mark(X3)) → U611(X1, X2, X3)
U121(X1, X2, active(X3)) → U121(X1, X2, X3)
ACTIVE(plus(N, 0)) → ISNAT(N)
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
PLUS(mark(X1), X2) → PLUS(X1, X2)
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
U631(X1, X2, mark(X3)) → U631(X1, X2, X3)
U411(mark(X)) → U411(X)
ACTIVE(isNatKind(plus(V1, V2))) → U311(isNatKind(V1), V2)
U411(active(X)) → U411(X)
MARK(U31(X1, X2)) → U311(mark(X1), X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
U211(X1, mark(X2)) → U211(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
U511(mark(X1), X2) → U511(X1, X2)
S(mark(X)) → S(X)
U151(active(X1), X2) → U151(X1, X2)
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U61(tt, M, N)) → U621(isNatKind(M), M, N)
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
U151(X1, active(X2)) → U151(X1, X2)
ACTIVE(U62(tt, M, N)) → U631(isNat(N), M, N)
U121(X1, X2, mark(X3)) → U121(X1, X2, X3)
U221(X1, active(X2)) → U221(X1, X2)
U231(mark(X)) → U231(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2, X3)) → U111(mark(X1), X2, X3)
U641(X1, active(X2), X3) → U641(X1, X2, X3)
U151(X1, mark(X2)) → U151(X1, X2)
ACTIVE(U41(tt)) → MARK(tt)
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(plus(N, s(M))) → U611(isNat(M), M, N)
U161(mark(X)) → U161(X)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U51(X1, X2)) → MARK(X1)
U631(X1, mark(X2), X3) → U631(X1, X2, X3)
U311(X1, active(X2)) → U311(X1, X2)
S(active(X)) → S(X)
MARK(U11(X1, X2, X3)) → MARK(X1)
U321(active(X)) → U321(X)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
MARK(U13(X1, X2, X3)) → MARK(X1)
ACTIVE(U62(tt, M, N)) → ISNAT(N)
U121(active(X1), X2, X3) → U121(X1, X2, X3)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
U511(active(X1), X2) → U511(X1, X2)
U631(active(X1), X2, X3) → U631(X1, X2, X3)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U14(tt, V1, V2)) → ISNAT(V1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
U621(X1, mark(X2), X3) → U621(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
PLUS(active(X1), X2) → PLUS(X1, X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U11(tt, V1, V2)) → ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → MARK(X1)
U521(X1, active(X2)) → U521(X1, X2)
ACTIVE(U64(tt, M, N)) → S(plus(N, M))
U511(X1, mark(X2)) → U511(X1, X2)
U621(X1, active(X2), X3) → U621(X1, X2, X3)
ACTIVE(plus(N, 0)) → U511(isNat(N), N)
ACTIVE(U64(tt, M, N)) → PLUS(N, M)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)
ACTIVE(U22(tt, V1)) → U231(isNat(V1))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
U621(active(X1), X2, X3) → U621(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
ISNAT(active(X)) → ISNAT(X)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → U161(isNat(V2))
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U641(X1, X2, active(X3)) → U641(X1, X2, X3)
ACTIVE(U51(tt, N)) → U521(isNatKind(N), N)
MARK(U23(X)) → MARK(X)
U621(mark(X1), X2, X3) → U621(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
ACTIVE(U11(tt, V1, V2)) → U121(isNatKind(V1), V1, V2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U61(X1, X2, X3)) → U611(mark(X1), X2, X3)
ISNATKIND(active(X)) → ISNATKIND(X)
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U13(tt, V1, V2)) → U141(isNatKind(V2), V1, V2)
ACTIVE(U32(tt)) → MARK(tt)
U311(active(X1), X2) → U311(X1, X2)
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U641(mark(X1), X2, X3) → U641(X1, X2, X3)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U211(mark(X1), X2) → U211(X1, X2)
U231(active(X)) → U231(X)
MARK(U12(X1, X2, X3)) → U121(mark(X1), X2, X3)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
U631(mark(X1), X2, X3) → U631(X1, X2, X3)
U521(active(X1), X2) → U521(X1, X2)
ACTIVE(U16(tt)) → MARK(tt)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
MARK(U62(X1, X2, X3)) → U621(mark(X1), X2, X3)
ACTIVE(isNat(plus(V1, V2))) → U111(isNatKind(V1), V1, V2)
U321(mark(X)) → U321(X)
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
U121(mark(X1), X2, X3) → U121(X1, X2, X3)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(U21(tt, V1)) → ISNATKIND(V1)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V1)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
ACTIVE(U14(tt, V1, V2)) → U151(isNat(V1), V2)
ACTIVE(U31(tt, V2)) → ISNATKIND(V2)
U211(active(X1), X2) → U211(X1, X2)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(plus(X1, X2)) → MARK(X1)
ISNAT(mark(X)) → ISNAT(X)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U611(X1, X2, active(X3)) → U611(X1, X2, X3)
MARK(U16(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
U221(X1, mark(X2)) → U221(X1, X2)
MARK(tt) → ACTIVE(tt)
ACTIVE(isNatKind(s(V1))) → U411(isNatKind(V1))
U631(X1, X2, active(X3)) → U631(X1, X2, X3)
MARK(U41(X)) → U411(mark(X))
MARK(U32(X)) → U321(mark(X))
ACTIVE(U21(tt, V1)) → U221(isNatKind(V1), V1)
MARK(U22(X1, X2)) → U221(mark(X1), X2)
U121(X1, active(X2), X3) → U121(X1, X2, X3)
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U63(tt, M, N)) → U641(isNatKind(N), M, N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U16(X)) → U161(mark(X))
MARK(U21(X1, X2)) → U211(mark(X1), X2)
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
U221(mark(X1), X2) → U221(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
U521(X1, mark(X2)) → U521(X1, X2)
U641(active(X1), X2, X3) → U641(X1, X2, X3)
ACTIVE(U31(tt, V2)) → U321(isNatKind(V2))
ACTIVE(U12(tt, V1, V2)) → ISNATKIND(V2)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
U621(X1, X2, mark(X3)) → U621(X1, X2, X3)
ACTIVE(U15(tt, V2)) → ISNAT(V2)
U111(X1, mark(X2), X3) → U111(X1, X2, X3)
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U13(tt, V1, V2)) → ISNATKIND(V2)
U221(active(X1), X2) → U221(X1, X2)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U521(mark(X1), X2) → U521(X1, X2)
MARK(s(X)) → S(mark(X))
ISNATKIND(mark(X)) → ISNATKIND(X)
U621(X1, X2, active(X3)) → U621(X1, X2, X3)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
U611(X1, active(X2), X3) → U611(X1, X2, X3)
MARK(U51(X1, X2)) → U511(mark(X1), X2)
ACTIVE(isNatKind(0)) → MARK(tt)
MARK(U52(X1, X2)) → U521(mark(X1), X2)
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U63(X1, X2, X3)) → U631(mark(X1), X2, X3)
MARK(U23(X)) → U231(mark(X))
MARK(0) → ACTIVE(0)
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
U511(X1, active(X2)) → U511(X1, X2)
U611(X1, mark(X2), X3) → U611(X1, X2, X3)
MARK(U41(X)) → MARK(X)
U611(active(X1), X2, X3) → U611(X1, X2, X3)
MARK(U14(X1, X2, X3)) → U141(mark(X1), X2, X3)
MARK(U64(X1, X2, X3)) → U641(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.

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U161(active(X)) → U161(X)
U641(X1, mark(X2), X3) → U641(X1, X2, X3)
U121(X1, mark(X2), X3) → U121(X1, X2, X3)
ACTIVE(U63(tt, M, N)) → ISNATKIND(N)
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
U631(X1, active(X2), X3) → U631(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
ACTIVE(U61(tt, M, N)) → ISNATKIND(M)
MARK(U15(X1, X2)) → U151(mark(X1), X2)
ACTIVE(U22(tt, V1)) → ISNAT(V1)
U151(mark(X1), X2) → U151(X1, X2)
U311(mark(X1), X2) → U311(X1, X2)
U311(X1, mark(X2)) → U311(X1, X2)
MARK(U32(X)) → MARK(X)
MARK(U13(X1, X2, X3)) → U131(mark(X1), X2, X3)
ACTIVE(U12(tt, V1, V2)) → U131(isNatKind(V2), V1, V2)
ACTIVE(U51(tt, N)) → ISNATKIND(N)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
U641(X1, X2, mark(X3)) → U641(X1, X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → MARK(X1)
U611(X1, X2, mark(X3)) → U611(X1, X2, X3)
U121(X1, X2, active(X3)) → U121(X1, X2, X3)
ACTIVE(plus(N, 0)) → ISNAT(N)
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
PLUS(mark(X1), X2) → PLUS(X1, X2)
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
U631(X1, X2, mark(X3)) → U631(X1, X2, X3)
U411(mark(X)) → U411(X)
ACTIVE(isNatKind(plus(V1, V2))) → U311(isNatKind(V1), V2)
U411(active(X)) → U411(X)
MARK(U31(X1, X2)) → U311(mark(X1), X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
U211(X1, mark(X2)) → U211(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
U511(mark(X1), X2) → U511(X1, X2)
S(mark(X)) → S(X)
U151(active(X1), X2) → U151(X1, X2)
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U61(tt, M, N)) → U621(isNatKind(M), M, N)
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
U151(X1, active(X2)) → U151(X1, X2)
ACTIVE(U62(tt, M, N)) → U631(isNat(N), M, N)
U121(X1, X2, mark(X3)) → U121(X1, X2, X3)
U221(X1, active(X2)) → U221(X1, X2)
U231(mark(X)) → U231(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2, X3)) → U111(mark(X1), X2, X3)
U641(X1, active(X2), X3) → U641(X1, X2, X3)
U151(X1, mark(X2)) → U151(X1, X2)
ACTIVE(U41(tt)) → MARK(tt)
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(plus(N, s(M))) → U611(isNat(M), M, N)
U161(mark(X)) → U161(X)
U131(X1, X2, active(X3)) → U131(X1, X2, X3)
MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U51(X1, X2)) → MARK(X1)
U631(X1, mark(X2), X3) → U631(X1, X2, X3)
U311(X1, active(X2)) → U311(X1, X2)
S(active(X)) → S(X)
MARK(U11(X1, X2, X3)) → MARK(X1)
U321(active(X)) → U321(X)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
MARK(U13(X1, X2, X3)) → MARK(X1)
ACTIVE(U62(tt, M, N)) → ISNAT(N)
U121(active(X1), X2, X3) → U121(X1, X2, X3)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
U511(active(X1), X2) → U511(X1, X2)
U631(active(X1), X2, X3) → U631(X1, X2, X3)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U14(tt, V1, V2)) → ISNAT(V1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
U621(X1, mark(X2), X3) → U621(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
PLUS(active(X1), X2) → PLUS(X1, X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U11(tt, V1, V2)) → ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → MARK(X1)
U521(X1, active(X2)) → U521(X1, X2)
ACTIVE(U64(tt, M, N)) → S(plus(N, M))
U511(X1, mark(X2)) → U511(X1, X2)
U621(X1, active(X2), X3) → U621(X1, X2, X3)
ACTIVE(plus(N, 0)) → U511(isNat(N), N)
ACTIVE(U64(tt, M, N)) → PLUS(N, M)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)
ACTIVE(U22(tt, V1)) → U231(isNat(V1))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
U621(active(X1), X2, X3) → U621(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
ISNAT(active(X)) → ISNAT(X)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(U15(tt, V2)) → U161(isNat(V2))
U131(X1, mark(X2), X3) → U131(X1, X2, X3)
U641(X1, X2, active(X3)) → U641(X1, X2, X3)
ACTIVE(U51(tt, N)) → U521(isNatKind(N), N)
MARK(U23(X)) → MARK(X)
U621(mark(X1), X2, X3) → U621(X1, X2, X3)
U131(active(X1), X2, X3) → U131(X1, X2, X3)
ACTIVE(U11(tt, V1, V2)) → U121(isNatKind(V1), V1, V2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U62(X1, X2, X3)) → MARK(X1)
MARK(U61(X1, X2, X3)) → U611(mark(X1), X2, X3)
ISNATKIND(active(X)) → ISNATKIND(X)
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U13(tt, V1, V2)) → U141(isNatKind(V2), V1, V2)
ACTIVE(U32(tt)) → MARK(tt)
U311(active(X1), X2) → U311(X1, X2)
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
U141(X1, X2, active(X3)) → U141(X1, X2, X3)
MARK(U63(X1, X2, X3)) → MARK(X1)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U641(mark(X1), X2, X3) → U641(X1, X2, X3)
U141(mark(X1), X2, X3) → U141(X1, X2, X3)
U211(mark(X1), X2) → U211(X1, X2)
U231(active(X)) → U231(X)
MARK(U12(X1, X2, X3)) → U121(mark(X1), X2, X3)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
U631(mark(X1), X2, X3) → U631(X1, X2, X3)
U521(active(X1), X2) → U521(X1, X2)
ACTIVE(U16(tt)) → MARK(tt)
U131(X1, X2, mark(X3)) → U131(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
MARK(U62(X1, X2, X3)) → U621(mark(X1), X2, X3)
ACTIVE(isNat(plus(V1, V2))) → U111(isNatKind(V1), V1, V2)
U321(mark(X)) → U321(X)
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
U121(mark(X1), X2, X3) → U121(X1, X2, X3)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(U21(tt, V1)) → ISNATKIND(V1)
U131(mark(X1), X2, X3) → U131(X1, X2, X3)
ACTIVE(isNat(plus(V1, V2))) → ISNATKIND(V1)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
ACTIVE(U14(tt, V1, V2)) → U151(isNat(V1), V2)
ACTIVE(U31(tt, V2)) → ISNATKIND(V2)
U211(active(X1), X2) → U211(X1, X2)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(plus(X1, X2)) → MARK(X1)
ISNAT(mark(X)) → ISNAT(X)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
U131(X1, active(X2), X3) → U131(X1, X2, X3)
U611(X1, X2, active(X3)) → U611(X1, X2, X3)
MARK(U16(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
U221(X1, mark(X2)) → U221(X1, X2)
MARK(tt) → ACTIVE(tt)
ACTIVE(isNatKind(s(V1))) → U411(isNatKind(V1))
U631(X1, X2, active(X3)) → U631(X1, X2, X3)
MARK(U41(X)) → U411(mark(X))
MARK(U32(X)) → U321(mark(X))
ACTIVE(U21(tt, V1)) → U221(isNatKind(V1), V1)
MARK(U22(X1, X2)) → U221(mark(X1), X2)
U121(X1, active(X2), X3) → U121(X1, X2, X3)
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U63(tt, M, N)) → U641(isNatKind(N), M, N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U16(X)) → U161(mark(X))
MARK(U21(X1, X2)) → U211(mark(X1), X2)
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
U221(mark(X1), X2) → U221(X1, X2)
U211(X1, active(X2)) → U211(X1, X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
U521(X1, mark(X2)) → U521(X1, X2)
U641(active(X1), X2, X3) → U641(X1, X2, X3)
ACTIVE(U31(tt, V2)) → U321(isNatKind(V2))
ACTIVE(U12(tt, V1, V2)) → ISNATKIND(V2)
ACTIVE(isNatKind(plus(V1, V2))) → ISNATKIND(V1)
U621(X1, X2, mark(X3)) → U621(X1, X2, X3)
ACTIVE(U15(tt, V2)) → ISNAT(V2)
U111(X1, mark(X2), X3) → U111(X1, X2, X3)
MARK(U16(X)) → ACTIVE(U16(mark(X)))
ACTIVE(U13(tt, V1, V2)) → ISNATKIND(V2)
U221(active(X1), X2) → U221(X1, X2)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U521(mark(X1), X2) → U521(X1, X2)
MARK(s(X)) → S(mark(X))
ISNATKIND(mark(X)) → ISNATKIND(X)
U621(X1, X2, active(X3)) → U621(X1, X2, X3)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
U611(X1, active(X2), X3) → U611(X1, X2, X3)
MARK(U51(X1, X2)) → U511(mark(X1), X2)
ACTIVE(isNatKind(0)) → MARK(tt)
MARK(U52(X1, X2)) → U521(mark(X1), X2)
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U63(X1, X2, X3)) → U631(mark(X1), X2, X3)
MARK(U23(X)) → U231(mark(X))
MARK(0) → ACTIVE(0)
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
U511(X1, active(X2)) → U511(X1, X2)
U611(X1, mark(X2), X3) → U611(X1, X2, X3)
MARK(U41(X)) → MARK(X)
U611(active(X1), X2, X3) → U611(X1, X2, X3)
MARK(U14(X1, X2, X3)) → U141(mark(X1), X2, X3)
MARK(U64(X1, X2, X3)) → U641(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.
The approximation of the Dependency Graph [15,17,22] contains 23 SCCs with 66 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S(mark(X)) → S(X)
S(active(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U641(X1, X2, active(X3)) → U641(X1, X2, X3)
U641(active(X1), X2, X3) → U641(X1, X2, X3)
U641(X1, mark(X2), X3) → U641(X1, X2, X3)
U641(X1, active(X2), X3) → U641(X1, X2, X3)
U641(X1, X2, mark(X3)) → U641(X1, X2, X3)
U641(mark(X1), X2, 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U631(mark(X1), X2, X3) → U631(X1, X2, X3)
U631(X1, X2, active(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, mark(X3)) → U631(X1, X2, X3)
U631(X1, mark(X2), 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U621(X1, active(X2), X3) → U621(X1, X2, X3)
U621(active(X1), X2, X3) → U621(X1, X2, X3)
U621(mark(X1), X2, X3) → U621(X1, X2, X3)
U621(X1, X2, active(X3)) → U621(X1, X2, X3)
U621(X1, mark(X2), X3) → U621(X1, X2, X3)
U621(X1, X2, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U611(mark(X1), X2, X3) → U611(X1, X2, X3)
U611(X1, X2, active(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, mark(X2), X3) → U611(X1, X2, X3)
U611(active(X1), X2, 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U521(active(X1), X2) → U521(X1, X2)
U521(mark(X1), X2) → U521(X1, X2)
U521(X1, active(X2)) → U521(X1, X2)
U521(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U511(X1, mark(X2)) → U511(X1, X2)
U511(active(X1), X2) → U511(X1, X2)
U511(X1, active(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U311(X1, active(X2)) → U311(X1, X2)
U311(active(X1), X2) → U311(X1, X2)
U311(mark(X1), X2) → U311(X1, X2)
U311(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U231(mark(X)) → U231(X)
U231(active(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U221(mark(X1), X2) → U221(X1, X2)
U221(X1, active(X2)) → U221(X1, X2)
U221(active(X1), X2) → U221(X1, X2)
U221(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U211(X1, active(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U151(mark(X1), X2) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U151(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U141(X1, X2, active(X3)) → U141(X1, X2, X3)
U141(active(X1), X2, X3) → U141(X1, X2, X3)
U141(X1, X2, mark(X3)) → U141(X1, X2, X3)
U141(X1, mark(X2), X3) → U141(X1, X2, X3)
U141(X1, active(X2), X3) → U141(X1, X2, X3)
U141(mark(X1), X2, 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP

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

U121(X1, X2, mark(X3)) → U121(X1, X2, X3)
U121(active(X1), X2, X3) → U121(X1, X2, X3)
U121(X1, X2, active(X3)) → U121(X1, X2, X3)
U121(X1, mark(X2), X3) → U121(X1, X2, X3)
U121(mark(X1), X2, X3) → U121(X1, X2, X3)
U121(X1, active(X2), 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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ UsableRulesProof
          ↳ QDP

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

U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
U111(X1, mark(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
QDP
                ↳ QDPSizeChangeProof
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U41(X)) → ACTIVE(U41(mark(X)))
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(U16(X)) → ACTIVE(U16(mark(X)))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U61(X1, X2, X3)) → MARK(X1)
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U62(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 1   
POL(U11(x1, x2, x3)) = 1   
POL(U12(x1, x2, x3)) = 1   
POL(U13(x1, x2, x3)) = 1   
POL(U14(x1, x2, x3)) = 1   
POL(U15(x1, x2)) = 1   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 1   
POL(U22(x1, x2)) = 1   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 1   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 1   
POL(U52(x1, x2)) = 1   
POL(U61(x1, x2, x3)) = 1   
POL(U62(x1, x2, x3)) = 1   
POL(U63(x1, x2, x3)) = 1   
POL(U64(x1, x2, x3)) = 1   
POL(active(x1)) = 0   
POL(isNat(x1)) = 1   
POL(isNatKind(x1)) = 1   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 1   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(plus(N, 0)) → MARK(U51(isNat(N), N))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U61(X1, X2, X3)) → MARK(X1)
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U62(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1, x2)) = x1 + x2   
POL(U61(x1, x2, x3)) = x1 + x2 + x3   
POL(U62(x1, x2, x3)) = x1 + x2 + x3   
POL(U63(x1, x2, x3)) = x1 + x2 + x3   
POL(U64(x1, x2, x3)) = x1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U51(X1, X2)) → MARK(X1)
ACTIVE(U51(tt, N)) → MARK(U52(isNatKind(N), N))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U52(tt, N)) → MARK(N)
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U62(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 1 + x1 + x2   
POL(U52(x1, x2)) = x1 + x2   
POL(U61(x1, x2, x3)) = x1 + x2 + x3   
POL(U62(x1, x2, x3)) = x1 + x2 + x3   
POL(U63(x1, x2, x3)) = x1 + x2 + x3   
POL(U64(x1, x2, x3)) = x1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U52(X1, X2)) → MARK(X1)
ACTIVE(U52(tt, N)) → MARK(N)
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U62(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 1 + x2   
POL(U52(x1, x2)) = 1 + x1 + x2   
POL(U61(x1, x2, x3)) = x1 + x2 + x3   
POL(U62(x1, x2, x3)) = x1 + x2 + x3   
POL(U63(x1, x2, x3)) = x1 + x2 + x3   
POL(U64(x1, x2, x3)) = x1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
QDP
                            ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(plus(X1, X2)) → MARK(X2)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X1)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U62(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 1   
POL(U11(x1, x2, x3)) = 1   
POL(U12(x1, x2, x3)) = 1   
POL(U13(x1, x2, x3)) = 1   
POL(U14(x1, x2, x3)) = 1   
POL(U15(x1, x2)) = 1   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 1   
POL(U22(x1, x2)) = 1   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 1   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 1   
POL(U62(x1, x2, x3)) = 1   
POL(U63(x1, x2, x3)) = 1   
POL(U64(x1, x2, x3)) = 1   
POL(active(x1)) = 0   
POL(isNat(x1)) = 1   
POL(isNatKind(x1)) = 1   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 1   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
plus(mark(X1), X2) → plus(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U63(active(X1), X2, X3) → U63(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(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(X1, X2, active(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U51(X1, mark(X2)) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
QDP
                                ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(plus(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U61(X1, X2, X3)) → MARK(X1)
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U62(X1, X2, X3)) → MARK(X1)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1, x2)) = x2   
POL(U61(x1, x2, x3)) = x1 + x2 + x3   
POL(U62(x1, x2, x3)) = x1 + x2 + x3   
POL(U63(x1, x2, x3)) = x1 + x2 + x3   
POL(U64(x1, x2, x3)) = x1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 1   
POL(isNatKind(x1)) = 1   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 1   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
QDP
                                    ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U63(X1, X2, X3)) → MARK(X1)
MARK(U64(X1, X2, X3)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U62(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → MARK(X1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 1 + x2   
POL(U52(x1, x2)) = 1 + x2   
POL(U61(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U62(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U63(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U64(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 1 + x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
QDP
                                        ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


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.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(X)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 1   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
QDP
                                            ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(plus(N, s(M))) → MARK(U61(isNat(M), M, N))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U61(X1, X2, X3)) → ACTIVE(U61(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 1   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U63(active(X1), X2, X3) → U63(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(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(X1, X2, active(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U62(X1, X2, mark(X3)) → U62(X1, X2, X3)
U62(X1, active(X2), X3) → U62(X1, X2, X3)
U62(X1, mark(X2), X3) → U62(X1, X2, X3)
U62(active(X1), X2, X3) → U62(X1, X2, X3)
U62(X1, X2, active(X3)) → U62(X1, X2, X3)
U62(mark(X1), X2, X3) → U62(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U61(X1, X2, active(X3)) → U61(X1, X2, X3)
U61(X1, active(X2), X3) → U61(X1, X2, X3)
U61(X1, mark(X2), X3) → U61(X1, X2, X3)
U61(mark(X1), X2, X3) → U61(X1, X2, X3)
U61(active(X1), X2, X3) → U61(X1, X2, X3)
U61(X1, X2, mark(X3)) → U61(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
QDP
                                                ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


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.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(X)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 1 + x2 + x3   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
QDP
                                                    ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
ACTIVE(U61(tt, M, N)) → MARK(U62(isNatKind(M), M, N))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


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.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(U62(X1, X2, X3)) → ACTIVE(U62(mark(X1), X2, X3))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = x1 + x2 + x3   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
QDP
                                                        ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


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.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(X)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 1   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
QDP
                                                            ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U62(tt, M, N)) → MARK(U63(isNat(N), M, N))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U63(X1, X2, X3)) → ACTIVE(U63(mark(X1), X2, X3))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 1 + x1   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U63(active(X1), X2, X3) → U63(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(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(X1, X2, active(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
QDP
                                                                ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


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.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(X)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 1   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
QDP
                                                                    ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U63(tt, M, N)) → MARK(U64(isNatKind(N), M, N))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U64(X1, X2, X3)) → ACTIVE(U64(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 1   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U64(X1, mark(X2), X3) → U64(X1, X2, X3)
U64(X1, X2, active(X3)) → U64(X1, X2, X3)
U64(active(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, active(X2), X3) → U64(X1, X2, X3)
U64(mark(X1), X2, X3) → U64(X1, X2, X3)
U64(X1, X2, mark(X3)) → U64(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
QDP
                                                                        ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].


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.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(X)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 1   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
QDP
                                                                            ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U64(tt, M, N)) → MARK(s(plus(N, M)))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = x1 + x2 + x3   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
QDP
                                                                                ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(U13(x1, x2, x3)) = x1 + x2 + x3   
POL(U14(x1, x2, x3)) = x1 + x2 + x3   
POL(U15(x1, x2)) = x1 + x2   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1 + x2   
POL(U22(x1, x2)) = x1 + x2   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = x2   
POL(U52(x1, x2)) = x2   
POL(U61(x1, x2, x3)) = 1 + x2 + x3   
POL(U62(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U63(x1, x2, x3)) = 1 + x2 + x3   
POL(U64(x1, x2, x3)) = 1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = x1   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(s(x1)) = 1 + x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
QDP
                                                                                    ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U21(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 1 + x1 + x2   
POL(U22(x1, x2)) = x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
QDP
                                                                                        ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = x1 + x2   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
QDP
                                                                                            ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 1 + x1   
POL(U23(x1)) = x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
QDP
                                                                                                ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 1   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
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)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
QDP
                                                                                                    ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U23(X)) → MARK(X)
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U23(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x2   
POL(U22(x1, x2)) = x2   
POL(U23(x1)) = 1 + x1   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = x3   
POL(U62(x1, x2, x3)) = x3   
POL(U63(x1, x2, x3)) = x2   
POL(U64(x1, x2, x3)) = x1 + x2   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                        ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(isNat(plus(V1, V2))) → MARK(U11(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = x1 + x2 + x3   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(U13(x1, x2, x3)) = x1 + x2 + x3   
POL(U14(x1, x2, x3)) = x1 + x2 + x3   
POL(U15(x1, x2)) = x1 + x2   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = x2   
POL(U52(x1, x2)) = x2   
POL(U61(x1, x2, x3)) = x3   
POL(U62(x1, x2, x3)) = x1 + x3   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = x1   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
QDP
                                                                                                            ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 1 + x1   
POL(U12(x1, x2, x3)) = x1   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = x2   
POL(U62(x1, x2, x3)) = x3   
POL(U63(x1, x2, x3)) = x2 + x3   
POL(U64(x1, x2, x3)) = x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U11(tt, V1, V2)) → MARK(U12(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 1   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                    ↳ QDPOrderProof

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

MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U13(x1, x2, x3)) = x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                                        ↳ QDPOrderProof

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

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U12(tt, V1, V2)) → MARK(U13(isNatKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = x1 + x2 + x3   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
U13(X1, X2, active(X3)) → U13(X1, X2, X3)
U13(mark(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, mark(X2), X3) → U13(X1, X2, X3)
U13(active(X1), X2, X3) → U13(X1, X2, X3)
U13(X1, active(X2), X3) → U13(X1, X2, X3)
U13(X1, X2, mark(X3)) → U13(X1, X2, X3)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
QDP
                                                                                                                            ↳ QDPOrderProof

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

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U13(X1, X2, X3)) → ACTIVE(U13(mark(X1), X2, X3))
MARK(U13(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 1 + x1   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                                ↳ QDPOrderProof

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

MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U13(tt, V1, V2)) → MARK(U14(isNatKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = x1 + x3   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                                    ↳ QDPOrderProof

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

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(isNat(X)) → ACTIVE(isNat(X))
The remaining pairs can at least be oriented weakly.

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 1   
POL(U15(x1, x2)) = 1   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 1   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 1   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

U14(X1, mark(X2), X3) → U14(X1, X2, X3)
U14(active(X1), X2, X3) → U14(X1, X2, X3)
U14(X1, X2, mark(X3)) → U14(X1, X2, X3)
U14(mark(X1), 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(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                                                        ↳ QDPOrderProof

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

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U14(X1, X2, X3)) → MARK(X1)
MARK(U14(X1, X2, X3)) → ACTIVE(U14(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 1 + x1 + x2   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U15(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
QDP
                                                                                                                                            ↳ QDPOrderProof

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

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U14(tt, V1, V2)) → MARK(U15(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U41(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = x1   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

U15(active(X1), X2) → U15(X1, X2)
U15(mark(X1), X2) → U15(X1, X2)
U15(X1, mark(X2)) → U15(X1, X2)
U15(X1, active(X2)) → U15(X1, X2)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                                                ↳ QDPOrderProof

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

MARK(U16(X)) → MARK(X)
ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(isNatKind(plus(V1, V2))) → MARK(U31(isNatKind(V1), V2))
The remaining pairs can at least be oriented weakly.

MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U15(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1 + x2   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 1 + x2   
POL(U52(x1, x2)) = 1 + x2   
POL(U61(x1, x2, x3)) = 1 + x2 + x3   
POL(U62(x1, x2, x3)) = 1 + x2 + x3   
POL(U63(x1, x2, x3)) = 1 + x2 + x3   
POL(U64(x1, x2, x3)) = 1 + x2 + x3   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = x1   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                                                    ↳ QDPOrderProof

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

MARK(U16(X)) → MARK(X)
MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U15(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U15(X1, X2)) → ACTIVE(U15(mark(X1), X2))
MARK(U15(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U16(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 1 + x1   
POL(U16(x1)) = x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1 + x2   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
U16(mark(X)) → U16(X)
U16(active(X)) → U16(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                                                                        ↳ QDPOrderProof

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

MARK(U16(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U15(tt, V2)) → MARK(U16(isNat(V2)))
The remaining pairs can at least be oriented weakly.

MARK(U16(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 0   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = x1   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = 0   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
QDP
                                                                                                                                                            ↳ QDPOrderProof

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

MARK(U16(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U16(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 1 + x1   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = x1 + x2   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = x1   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                                                                ↳ QDPOrderProof

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

MARK(U31(X1, X2)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
The remaining pairs can at least be oriented weakly.

ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2, x3)) = 0   
POL(U12(x1, x2, x3)) = 0   
POL(U13(x1, x2, x3)) = 0   
POL(U14(x1, x2, x3)) = 0   
POL(U15(x1, x2)) = 0   
POL(U16(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U31(x1, x2)) = 1 + x1   
POL(U32(x1)) = x1   
POL(U41(x1)) = x1   
POL(U51(x1, x2)) = 0   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U62(x1, x2, x3)) = 0   
POL(U63(x1, x2, x3)) = 0   
POL(U64(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(mark(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                                                                    ↳ UsableRulesProof

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

ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ UsableRulesProof
QDP
                                                                                                                                                                        ↳ DependencyGraphProof

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

ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U31(tt, V2)) → MARK(U32(isNatKind(V2)))

The TRS R consists of the following rules:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ DependencyGraphProof
QDP
                                                                                                                                                                            ↳ UsableRulesProof

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

ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U41(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))

The TRS R consists of the following rules:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ UsableRulesProof
QDP
                                                                                                                                                                                ↳ UsableRulesReductionPairsProof

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

ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U32(X)) → MARK(X)
MARK(U41(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))

The TRS R consists of the following rules:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

ACTIVE(isNatKind(s(V1))) → MARK(U41(isNatKind(V1)))
MARK(U32(X)) → MARK(X)
The following rules are removed from R:

isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)
U41(mark(X)) → U41(X)
U41(active(X)) → U41(X)
Used ordering: POLO with Polynomial interpretation [25]:

POL(ACTIVE(x1)) = x1   
POL(MARK(x1)) = 2·x1   
POL(U32(x1)) = 2·x1   
POL(U41(x1)) = x1   
POL(active(x1)) = 2·x1   
POL(isNatKind(x1)) = 2·x1   
POL(mark(x1)) = 2·x1   
POL(s(x1)) = 2·x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ UsableRulesReductionPairsProof
QDP
                                                                                                                                                                                    ↳ DependencyGraphProof

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

MARK(U41(X)) → MARK(X)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                                                                  ↳ QDP
                                                                    ↳ QDPOrderProof
                                                                      ↳ QDP
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof
                                                                                                      ↳ QDP
                                                                                                        ↳ QDPOrderProof
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ QDPOrderProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ QDPOrderProof
                                                                                                                          ↳ QDP
                                                                                                                            ↳ QDPOrderProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ QDPOrderProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ UsableRulesReductionPairsProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ DependencyGraphProof
QDP
                                                                                                                                                                                        ↳ QDPSizeChangeProof

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

MARK(U41(X)) → MARK(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: