Termination w.r.t. Q proof of /tmp/tmpThlqmA/PEANO_complete_noand

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:

PLUS(active(X1), X2) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PLUS(mark(X1), X2) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PLUS(X1, mark(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
PLUS(X1, active(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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)

↳ 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)

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

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

U64¹(X1, X2, active(X3)) → U64¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U64¹(active(X1), X2, X3) → U64¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U64¹(X1, mark(X2), X3) → U64¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U64¹(X1, active(X2), X3) → U64¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U64¹(X1, X2, mark(X3)) → U64¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U64¹(mark(X1), X2, X3) → U64¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ 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)

↳ 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)

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

U63¹(mark(X1), X2, X3) → U63¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U63¹(X1, X2, active(X3)) → U63¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U63¹(active(X1), X2, X3) → U63¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U63¹(X1, active(X2), X3) → U63¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U63¹(X1, X2, mark(X3)) → U63¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U63¹(X1, mark(X2), X3) → U63¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ 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)

↳ 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)

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

U62¹(X1, active(X2), X3) → U62¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U62¹(active(X1), X2, X3) → U62¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U62¹(mark(X1), X2, X3) → U62¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U62¹(X1, X2, active(X3)) → U62¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U62¹(X1, mark(X2), X3) → U62¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U62¹(X1, X2, mark(X3)) → U62¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

↳ 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)

↳ 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)

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

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U61¹(X1, X2, active(X3)) → U61¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U61¹(X1, X2, mark(X3)) → U61¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U61¹(X1, active(X2), X3) → U61¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U61¹(X1, mark(X2), X3) → U61¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U61¹(active(X1), X2, X3) → U61¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ 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)

↳ 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)

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

U52¹(active(X1), X2) → U52¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U52¹(mark(X1), X2) → U52¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U52¹(X1, active(X2)) → U52¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U52¹(X1, mark(X2)) → U52¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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)

↳ 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)

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

U51¹(X1, mark(X2)) → U51¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U51¹(active(X1), X2) → U51¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U51¹(X1, active(X2)) → U51¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U51¹(mark(X1), X2) → U51¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ 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)

↳ 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)

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

U41¹(active(X)) → U41¹(X)
The graph contains the following edges 1 > 1
U41¹(mark(X)) → U41¹(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

U32¹(active(X)) → U32¹(X)
The graph contains the following edges 1 > 1
U32¹(mark(X)) → U32¹(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

U31¹(X1, active(X2)) → U31¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U31¹(active(X1), X2) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U31¹(mark(X1), X2) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U31¹(X1, mark(X2)) → U31¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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)

↳ 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)

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

U23¹(mark(X)) → U23¹(X)
The graph contains the following edges 1 > 1
U23¹(active(X)) → U23¹(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

U22¹(mark(X1), X2) → U22¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U22¹(X1, active(X2)) → U22¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U22¹(active(X1), X2) → U22¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U22¹(X1, mark(X2)) → U22¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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)

↳ 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)

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

U21¹(X1, active(X2)) → U21¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U21¹(active(X1), X2) → U21¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U21¹(X1, mark(X2)) → U21¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U21¹(mark(X1), X2) → U21¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ 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)

↳ 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)

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

U16¹(active(X)) → U16¹(X)
The graph contains the following edges 1 > 1
U16¹(mark(X)) → U16¹(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

ISNAT(active(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
ISNAT(mark(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

U15¹(mark(X1), X2) → U15¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U15¹(active(X1), X2) → U15¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U15¹(X1, active(X2)) → U15¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U15¹(X1, mark(X2)) → U15¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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)

↳ 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)

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

U14¹(X1, X2, active(X3)) → U14¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U14¹(active(X1), X2, X3) → U14¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U14¹(X1, X2, mark(X3)) → U14¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U14¹(X1, mark(X2), X3) → U14¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U14¹(X1, active(X2), X3) → U14¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U14¹(mark(X1), X2, X3) → U14¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ 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)

↳ 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)

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

U13¹(X1, active(X2), X3) → U13¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U13¹(X1, mark(X2), X3) → U13¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U13¹(X1, X2, mark(X3)) → U13¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U13¹(mark(X1), X2, X3) → U13¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U13¹(active(X1), X2, X3) → U13¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U13¹(X1, X2, active(X3)) → U13¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

↳ 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)

↳ 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)

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

ISNATKIND(active(X)) → ISNATKIND(X)
The graph contains the following edges 1 > 1
ISNATKIND(mark(X)) → ISNATKIND(X)
The graph contains the following edges 1 > 1

↳ 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)

↳ 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)

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

U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

↳ 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)

↳ 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)

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3

↳ 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)

↳ 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)

↳ 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)

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

MARK(U41(X)) → MARK(X)
The graph contains the following edges 1 > 1