The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
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)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(isNatKind(z0), z0, z1))
active(U12(tt, z0, z1)) → mark(U13(isNatKind(z1), z0, z1))
active(U13(tt, z0, z1)) → mark(U14(isNatKind(z1), z0, z1))
active(U14(tt, z0, z1)) → mark(U15(isNat(z0), z1))
active(U15(tt, z0)) → mark(U16(isNat(z0)))
active(U16(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNatKind(z0), z0))
active(U22(tt, z0)) → mark(U23(isNat(z0)))
active(U23(tt)) → mark(tt)
active(U31(tt, z0)) → mark(U32(isNatKind(z0)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, z0)) → mark(U52(isNatKind(z0), z0))
active(U52(tt, z0)) → mark(z0)
active(U61(tt, z0, z1)) → mark(U62(isNatKind(z0), z0, z1))
active(U62(tt, z0, z1)) → mark(U63(isNat(z1), z0, z1))
active(U63(tt, z0, z1)) → mark(U64(isNatKind(z1), z0, z1))
active(U64(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNatKind(z0), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(U31(isNatKind(z0), z1))
active(isNatKind(s(z0))) → mark(U41(isNatKind(z0)))
active(plus(z0, 0)) → mark(U51(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U61(isNat(z1), z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1, z2)) → active(U12(mark(z0), z1, z2))
mark(isNatKind(z0)) → active(isNatKind(z0))
mark(U13(z0, z1, z2)) → active(U13(mark(z0), z1, z2))
mark(U14(z0, z1, z2)) → active(U14(mark(z0), z1, z2))
mark(U15(z0, z1)) → active(U15(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(U16(z0)) → active(U16(mark(z0)))
mark(U21(z0, z1)) → active(U21(mark(z0), z1))
mark(U22(z0, z1)) → active(U22(mark(z0), z1))
mark(U23(z0)) → active(U23(mark(z0)))
mark(U31(z0, z1)) → active(U31(mark(z0), z1))
mark(U32(z0)) → active(U32(mark(z0)))
mark(U41(z0)) → active(U41(mark(z0)))
mark(U51(z0, z1)) → active(U51(mark(z0), z1))
mark(U52(z0, z1)) → active(U52(mark(z0), z1))
mark(U61(z0, z1, z2)) → active(U61(mark(z0), z1, z2))
mark(U62(z0, z1, z2)) → active(U62(mark(z0), z1, z2))
mark(U63(z0, z1, z2)) → active(U63(mark(z0), z1, z2))
mark(U64(z0, z1, z2)) → active(U64(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(0) → active(0)
U11(mark(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, mark(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, mark(z2)) → U11(z0, z1, z2)
U11(active(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, active(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, active(z2)) → U11(z0, z1, z2)
U12(mark(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, mark(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, mark(z2)) → U12(z0, z1, z2)
U12(active(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, active(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, active(z2)) → U12(z0, z1, z2)
isNatKind(mark(z0)) → isNatKind(z0)
isNatKind(active(z0)) → isNatKind(z0)
U13(mark(z0), z1, z2) → U13(z0, z1, z2)
U13(z0, mark(z1), z2) → U13(z0, z1, z2)
U13(z0, z1, mark(z2)) → U13(z0, z1, z2)
U13(active(z0), z1, z2) → U13(z0, z1, z2)
U13(z0, active(z1), z2) → U13(z0, z1, z2)
U13(z0, z1, active(z2)) → U13(z0, z1, z2)
U14(mark(z0), z1, z2) → U14(z0, z1, z2)
U14(z0, mark(z1), z2) → U14(z0, z1, z2)
U14(z0, z1, mark(z2)) → U14(z0, z1, z2)
U14(active(z0), z1, z2) → U14(z0, z1, z2)
U14(z0, active(z1), z2) → U14(z0, z1, z2)
U14(z0, z1, active(z2)) → U14(z0, z1, z2)
U15(mark(z0), z1) → U15(z0, z1)
U15(z0, mark(z1)) → U15(z0, z1)
U15(active(z0), z1) → U15(z0, z1)
U15(z0, active(z1)) → U15(z0, z1)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U16(mark(z0)) → U16(z0)
U16(active(z0)) → U16(z0)
U21(mark(z0), z1) → U21(z0, z1)
U21(z0, mark(z1)) → U21(z0, z1)
U21(active(z0), z1) → U21(z0, z1)
U21(z0, active(z1)) → U21(z0, z1)
U22(mark(z0), z1) → U22(z0, z1)
U22(z0, mark(z1)) → U22(z0, z1)
U22(active(z0), z1) → U22(z0, z1)
U22(z0, active(z1)) → U22(z0, z1)
U23(mark(z0)) → U23(z0)
U23(active(z0)) → U23(z0)
U31(mark(z0), z1) → U31(z0, z1)
U31(z0, mark(z1)) → U31(z0, z1)
U31(active(z0), z1) → U31(z0, z1)
U31(z0, active(z1)) → U31(z0, z1)
U32(mark(z0)) → U32(z0)
U32(active(z0)) → U32(z0)
U41(mark(z0)) → U41(z0)
U41(active(z0)) → U41(z0)
U51(mark(z0), z1) → U51(z0, z1)
U51(z0, mark(z1)) → U51(z0, z1)
U51(active(z0), z1) → U51(z0, z1)
U51(z0, active(z1)) → U51(z0, z1)
U52(mark(z0), z1) → U52(z0, z1)
U52(z0, mark(z1)) → U52(z0, z1)
U52(active(z0), z1) → U52(z0, z1)
U52(z0, active(z1)) → U52(z0, z1)
U61(mark(z0), z1, z2) → U61(z0, z1, z2)
U61(z0, mark(z1), z2) → U61(z0, z1, z2)
U61(z0, z1, mark(z2)) → U61(z0, z1, z2)
U61(active(z0), z1, z2) → U61(z0, z1, z2)
U61(z0, active(z1), z2) → U61(z0, z1, z2)
U61(z0, z1, active(z2)) → U61(z0, z1, z2)
U62(mark(z0), z1, z2) → U62(z0, z1, z2)
U62(z0, mark(z1), z2) → U62(z0, z1, z2)
U62(z0, z1, mark(z2)) → U62(z0, z1, z2)
U62(active(z0), z1, z2) → U62(z0, z1, z2)
U62(z0, active(z1), z2) → U62(z0, z1, z2)
U62(z0, z1, active(z2)) → U62(z0, z1, z2)
U63(mark(z0), z1, z2) → U63(z0, z1, z2)
U63(z0, mark(z1), z2) → U63(z0, z1, z2)
U63(z0, z1, mark(z2)) → U63(z0, z1, z2)
U63(active(z0), z1, z2) → U63(z0, z1, z2)
U63(z0, active(z1), z2) → U63(z0, z1, z2)
U63(z0, z1, active(z2)) → U63(z0, z1, z2)
U64(mark(z0), z1, z2) → U64(z0, z1, z2)
U64(z0, mark(z1), z2) → U64(z0, z1, z2)
U64(z0, z1, mark(z2)) → U64(z0, z1, z2)
U64(active(z0), z1, z2) → U64(z0, z1, z2)
U64(z0, active(z1), z2) → U64(z0, z1, z2)
U64(z0, z1, active(z2)) → U64(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(z0, z1)

Tuples:

ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(isNatKind(z0), z0, z1)), U12'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(U12(tt, z0, z1)) → c1(MARK(U13(isNatKind(z1), z0, z1)), U13'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U13(tt, z0, z1)) → c2(MARK(U14(isNatKind(z1), z0, z1)), U14'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U14(tt, z0, z1)) → c3(MARK(U15(isNat(z0), z1)), U15'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U15(tt, z0)) → c4(MARK(U16(isNat(z0))), U16'(isNat(z0)), ISNAT(z0))
ACTIVE(U16(tt)) → c5(MARK(tt))
ACTIVE(U21(tt, z0)) → c6(MARK(U22(isNatKind(z0), z0)), U22'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(U22(tt, z0)) → c7(MARK(U23(isNat(z0))), U23'(isNat(z0)), ISNAT(z0))
ACTIVE(U23(tt)) → c8(MARK(tt))
ACTIVE(U31(tt, z0)) → c9(MARK(U32(isNatKind(z0))), U32'(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(U32(tt)) → c10(MARK(tt))
ACTIVE(U41(tt)) → c11(MARK(tt))
ACTIVE(U51(tt, z0)) → c12(MARK(U52(isNatKind(z0), z0)), U52'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(U52(tt, z0)) → c13(MARK(z0))
ACTIVE(U61(tt, z0, z1)) → c14(MARK(U62(isNatKind(z0), z0, z1)), U62'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(U62(tt, z0, z1)) → c15(MARK(U63(isNat(z1), z0, z1)), U63'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U63(tt, z0, z1)) → c16(MARK(U64(isNatKind(z1), z0, z1)), U64'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U64(tt, z0, z1)) → c17(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(0)) → c18(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c19(MARK(U11(isNatKind(z0), z0, z1)), U11'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c20(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c21(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c22(MARK(U31(isNatKind(z0), z1)), U31'(isNatKind(z0), z1), ISNATKIND(z0))
ACTIVE(isNatKind(s(z0))) → c23(MARK(U41(isNatKind(z0))), U41'(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c24(MARK(U51(isNat(z0), z0)), U51'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c25(MARK(U61(isNat(z1), z1, z0)), U61'(isNat(z1), z1, z0), ISNAT(z1))
MARK(U11(z0, z1, z2)) → c26(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c27(ACTIVE(tt))
MARK(U12(z0, z1, z2)) → c28(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(isNatKind(z0)) → c29(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
MARK(U13(z0, z1, z2)) → c30(ACTIVE(U13(mark(z0), z1, z2)), U13'(mark(z0), z1, z2), MARK(z0))
MARK(U14(z0, z1, z2)) → c31(ACTIVE(U14(mark(z0), z1, z2)), U14'(mark(z0), z1, z2), MARK(z0))
MARK(U15(z0, z1)) → c32(ACTIVE(U15(mark(z0), z1)), U15'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c33(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U16(z0)) → c34(ACTIVE(U16(mark(z0))), U16'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c35(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0, z1)) → c36(ACTIVE(U22(mark(z0), z1)), U22'(mark(z0), z1), MARK(z0))
MARK(U23(z0)) → c37(ACTIVE(U23(mark(z0))), U23'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c38(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U32(z0)) → c39(ACTIVE(U32(mark(z0))), U32'(mark(z0)), MARK(z0))
MARK(U41(z0)) → c40(ACTIVE(U41(mark(z0))), U41'(mark(z0)), MARK(z0))
MARK(U51(z0, z1)) → c41(ACTIVE(U51(mark(z0), z1)), U51'(mark(z0), z1), MARK(z0))
MARK(U52(z0, z1)) → c42(ACTIVE(U52(mark(z0), z1)), U52'(mark(z0), z1), MARK(z0))
MARK(U61(z0, z1, z2)) → c43(ACTIVE(U61(mark(z0), z1, z2)), U61'(mark(z0), z1, z2), MARK(z0))
MARK(U62(z0, z1, z2)) → c44(ACTIVE(U62(mark(z0), z1, z2)), U62'(mark(z0), z1, z2), MARK(z0))
MARK(U63(z0, z1, z2)) → c45(ACTIVE(U63(mark(z0), z1, z2)), U63'(mark(z0), z1, z2), MARK(z0))
MARK(U64(z0, z1, z2)) → c46(ACTIVE(U64(mark(z0), z1, z2)), U64'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c47(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c48(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c49(ACTIVE(0))
U11'(mark(z0), z1, z2) → c50(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c51(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c52(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c53(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c54(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c55(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c56(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c57(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c58(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c59(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c60(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c61(U12'(z0, z1, z2))
ISNATKIND(mark(z0)) → c62(ISNATKIND(z0))
ISNATKIND(active(z0)) → c63(ISNATKIND(z0))
U13'(mark(z0), z1, z2) → c64(U13'(z0, z1, z2))
U13'(z0, mark(z1), z2) → c65(U13'(z0, z1, z2))
U13'(z0, z1, mark(z2)) → c66(U13'(z0, z1, z2))
U13'(active(z0), z1, z2) → c67(U13'(z0, z1, z2))
U13'(z0, active(z1), z2) → c68(U13'(z0, z1, z2))
U13'(z0, z1, active(z2)) → c69(U13'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c70(U14'(z0, z1, z2))
U14'(z0, mark(z1), z2) → c71(U14'(z0, z1, z2))
U14'(z0, z1, mark(z2)) → c72(U14'(z0, z1, z2))
U14'(active(z0), z1, z2) → c73(U14'(z0, z1, z2))
U14'(z0, active(z1), z2) → c74(U14'(z0, z1, z2))
U14'(z0, z1, active(z2)) → c75(U14'(z0, z1, z2))
U15'(mark(z0), z1) → c76(U15'(z0, z1))
U15'(z0, mark(z1)) → c77(U15'(z0, z1))
U15'(active(z0), z1) → c78(U15'(z0, z1))
U15'(z0, active(z1)) → c79(U15'(z0, z1))
ISNAT(mark(z0)) → c80(ISNAT(z0))
ISNAT(active(z0)) → c81(ISNAT(z0))
U16'(mark(z0)) → c82(U16'(z0))
U16'(active(z0)) → c83(U16'(z0))
U21'(mark(z0), z1) → c84(U21'(z0, z1))
U21'(z0, mark(z1)) → c85(U21'(z0, z1))
U21'(active(z0), z1) → c86(U21'(z0, z1))
U21'(z0, active(z1)) → c87(U21'(z0, z1))
U22'(mark(z0), z1) → c88(U22'(z0, z1))
U22'(z0, mark(z1)) → c89(U22'(z0, z1))
U22'(active(z0), z1) → c90(U22'(z0, z1))
U22'(z0, active(z1)) → c91(U22'(z0, z1))
U23'(mark(z0)) → c92(U23'(z0))
U23'(active(z0)) → c93(U23'(z0))
U31'(mark(z0), z1) → c94(U31'(z0, z1))
U31'(z0, mark(z1)) → c95(U31'(z0, z1))
U31'(active(z0), z1) → c96(U31'(z0, z1))
U31'(z0, active(z1)) → c97(U31'(z0, z1))
U32'(mark(z0)) → c98(U32'(z0))
U32'(active(z0)) → c99(U32'(z0))
U41'(mark(z0)) → c100(U41'(z0))
U41'(active(z0)) → c101(U41'(z0))
U51'(mark(z0), z1) → c102(U51'(z0, z1))
U51'(z0, mark(z1)) → c103(U51'(z0, z1))
U51'(active(z0), z1) → c104(U51'(z0, z1))
U51'(z0, active(z1)) → c105(U51'(z0, z1))
U52'(mark(z0), z1) → c106(U52'(z0, z1))
U52'(z0, mark(z1)) → c107(U52'(z0, z1))
U52'(active(z0), z1) → c108(U52'(z0, z1))
U52'(z0, active(z1)) → c109(U52'(z0, z1))
U61'(mark(z0), z1, z2) → c110(U61'(z0, z1, z2))
U61'(z0, mark(z1), z2) → c111(U61'(z0, z1, z2))
U61'(z0, z1, mark(z2)) → c112(U61'(z0, z1, z2))
U61'(active(z0), z1, z2) → c113(U61'(z0, z1, z2))
U61'(z0, active(z1), z2) → c114(U61'(z0, z1, z2))
U61'(z0, z1, active(z2)) → c115(U61'(z0, z1, z2))
U62'(mark(z0), z1, z2) → c116(U62'(z0, z1, z2))
U62'(z0, mark(z1), z2) → c117(U62'(z0, z1, z2))
U62'(z0, z1, mark(z2)) → c118(U62'(z0, z1, z2))
U62'(active(z0), z1, z2) → c119(U62'(z0, z1, z2))
U62'(z0, active(z1), z2) → c120(U62'(z0, z1, z2))
U62'(z0, z1, active(z2)) → c121(U62'(z0, z1, z2))
U63'(mark(z0), z1, z2) → c122(U63'(z0, z1, z2))
U63'(z0, mark(z1), z2) → c123(U63'(z0, z1, z2))
U63'(z0, z1, mark(z2)) → c124(U63'(z0, z1, z2))
U63'(active(z0), z1, z2) → c125(U63'(z0, z1, z2))
U63'(z0, active(z1), z2) → c126(U63'(z0, z1, z2))
U63'(z0, z1, active(z2)) → c127(U63'(z0, z1, z2))
U64'(mark(z0), z1, z2) → c128(U64'(z0, z1, z2))
U64'(z0, mark(z1), z2) → c129(U64'(z0, z1, z2))
U64'(z0, z1, mark(z2)) → c130(U64'(z0, z1, z2))
U64'(active(z0), z1, z2) → c131(U64'(z0, z1, z2))
U64'(z0, active(z1), z2) → c132(U64'(z0, z1, z2))
U64'(z0, z1, active(z2)) → c133(U64'(z0, z1, z2))
S(mark(z0)) → c134(S(z0))
S(active(z0)) → c135(S(z0))
PLUS(mark(z0), z1) → c136(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c137(PLUS(z0, z1))
PLUS(active(z0), z1) → c138(PLUS(z0, z1))
PLUS(z0, active(z1)) → c139(PLUS(z0, z1))

S tuples:

ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(isNatKind(z0), z0, z1)), U12'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(U12(tt, z0, z1)) → c1(MARK(U13(isNatKind(z1), z0, z1)), U13'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U13(tt, z0, z1)) → c2(MARK(U14(isNatKind(z1), z0, z1)), U14'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U14(tt, z0, z1)) → c3(MARK(U15(isNat(z0), z1)), U15'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U15(tt, z0)) → c4(MARK(U16(isNat(z0))), U16'(isNat(z0)), ISNAT(z0))
ACTIVE(U16(tt)) → c5(MARK(tt))
ACTIVE(U21(tt, z0)) → c6(MARK(U22(isNatKind(z0), z0)), U22'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(U22(tt, z0)) → c7(MARK(U23(isNat(z0))), U23'(isNat(z0)), ISNAT(z0))
ACTIVE(U23(tt)) → c8(MARK(tt))
ACTIVE(U31(tt, z0)) → c9(MARK(U32(isNatKind(z0))), U32'(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(U32(tt)) → c10(MARK(tt))
ACTIVE(U41(tt)) → c11(MARK(tt))
ACTIVE(U51(tt, z0)) → c12(MARK(U52(isNatKind(z0), z0)), U52'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(U52(tt, z0)) → c13(MARK(z0))
ACTIVE(U61(tt, z0, z1)) → c14(MARK(U62(isNatKind(z0), z0, z1)), U62'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(U62(tt, z0, z1)) → c15(MARK(U63(isNat(z1), z0, z1)), U63'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U63(tt, z0, z1)) → c16(MARK(U64(isNatKind(z1), z0, z1)), U64'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U64(tt, z0, z1)) → c17(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(0)) → c18(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c19(MARK(U11(isNatKind(z0), z0, z1)), U11'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c20(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c21(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c22(MARK(U31(isNatKind(z0), z1)), U31'(isNatKind(z0), z1), ISNATKIND(z0))
ACTIVE(isNatKind(s(z0))) → c23(MARK(U41(isNatKind(z0))), U41'(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c24(MARK(U51(isNat(z0), z0)), U51'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c25(MARK(U61(isNat(z1), z1, z0)), U61'(isNat(z1), z1, z0), ISNAT(z1))
MARK(U11(z0, z1, z2)) → c26(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c27(ACTIVE(tt))
MARK(U12(z0, z1, z2)) → c28(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(isNatKind(z0)) → c29(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
MARK(U13(z0, z1, z2)) → c30(ACTIVE(U13(mark(z0), z1, z2)), U13'(mark(z0), z1, z2), MARK(z0))
MARK(U14(z0, z1, z2)) → c31(ACTIVE(U14(mark(z0), z1, z2)), U14'(mark(z0), z1, z2), MARK(z0))
MARK(U15(z0, z1)) → c32(ACTIVE(U15(mark(z0), z1)), U15'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c33(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U16(z0)) → c34(ACTIVE(U16(mark(z0))), U16'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c35(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0, z1)) → c36(ACTIVE(U22(mark(z0), z1)), U22'(mark(z0), z1), MARK(z0))
MARK(U23(z0)) → c37(ACTIVE(U23(mark(z0))), U23'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c38(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U32(z0)) → c39(ACTIVE(U32(mark(z0))), U32'(mark(z0)), MARK(z0))
MARK(U41(z0)) → c40(ACTIVE(U41(mark(z0))), U41'(mark(z0)), MARK(z0))
MARK(U51(z0, z1)) → c41(ACTIVE(U51(mark(z0), z1)), U51'(mark(z0), z1), MARK(z0))
MARK(U52(z0, z1)) → c42(ACTIVE(U52(mark(z0), z1)), U52'(mark(z0), z1), MARK(z0))
MARK(U61(z0, z1, z2)) → c43(ACTIVE(U61(mark(z0), z1, z2)), U61'(mark(z0), z1, z2), MARK(z0))
MARK(U62(z0, z1, z2)) → c44(ACTIVE(U62(mark(z0), z1, z2)), U62'(mark(z0), z1, z2), MARK(z0))
MARK(U63(z0, z1, z2)) → c45(ACTIVE(U63(mark(z0), z1, z2)), U63'(mark(z0), z1, z2), MARK(z0))
MARK(U64(z0, z1, z2)) → c46(ACTIVE(U64(mark(z0), z1, z2)), U64'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c47(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c48(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c49(ACTIVE(0))
U11'(mark(z0), z1, z2) → c50(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c51(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c52(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c53(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c54(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c55(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c56(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c57(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c58(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c59(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c60(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c61(U12'(z0, z1, z2))
ISNATKIND(mark(z0)) → c62(ISNATKIND(z0))
ISNATKIND(active(z0)) → c63(ISNATKIND(z0))
U13'(mark(z0), z1, z2) → c64(U13'(z0, z1, z2))
U13'(z0, mark(z1), z2) → c65(U13'(z0, z1, z2))
U13'(z0, z1, mark(z2)) → c66(U13'(z0, z1, z2))
U13'(active(z0), z1, z2) → c67(U13'(z0, z1, z2))
U13'(z0, active(z1), z2) → c68(U13'(z0, z1, z2))
U13'(z0, z1, active(z2)) → c69(U13'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c70(U14'(z0, z1, z2))
U14'(z0, mark(z1), z2) → c71(U14'(z0, z1, z2))
U14'(z0, z1, mark(z2)) → c72(U14'(z0, z1, z2))
U14'(active(z0), z1, z2) → c73(U14'(z0, z1, z2))
U14'(z0, active(z1), z2) → c74(U14'(z0, z1, z2))
U14'(z0, z1, active(z2)) → c75(U14'(z0, z1, z2))
U15'(mark(z0), z1) → c76(U15'(z0, z1))
U15'(z0, mark(z1)) → c77(U15'(z0, z1))
U15'(active(z0), z1) → c78(U15'(z0, z1))
U15'(z0, active(z1)) → c79(U15'(z0, z1))
ISNAT(mark(z0)) → c80(ISNAT(z0))
ISNAT(active(z0)) → c81(ISNAT(z0))
U16'(mark(z0)) → c82(U16'(z0))
U16'(active(z0)) → c83(U16'(z0))
U21'(mark(z0), z1) → c84(U21'(z0, z1))
U21'(z0, mark(z1)) → c85(U21'(z0, z1))
U21'(active(z0), z1) → c86(U21'(z0, z1))
U21'(z0, active(z1)) → c87(U21'(z0, z1))
U22'(mark(z0), z1) → c88(U22'(z0, z1))
U22'(z0, mark(z1)) → c89(U22'(z0, z1))
U22'(active(z0), z1) → c90(U22'(z0, z1))
U22'(z0, active(z1)) → c91(U22'(z0, z1))
U23'(mark(z0)) → c92(U23'(z0))
U23'(active(z0)) → c93(U23'(z0))
U31'(mark(z0), z1) → c94(U31'(z0, z1))
U31'(z0, mark(z1)) → c95(U31'(z0, z1))
U31'(active(z0), z1) → c96(U31'(z0, z1))
U31'(z0, active(z1)) → c97(U31'(z0, z1))
U32'(mark(z0)) → c98(U32'(z0))
U32'(active(z0)) → c99(U32'(z0))
U41'(mark(z0)) → c100(U41'(z0))
U41'(active(z0)) → c101(U41'(z0))
U51'(mark(z0), z1) → c102(U51'(z0, z1))
U51'(z0, mark(z1)) → c103(U51'(z0, z1))
U51'(active(z0), z1) → c104(U51'(z0, z1))
U51'(z0, active(z1)) → c105(U51'(z0, z1))
U52'(mark(z0), z1) → c106(U52'(z0, z1))
U52'(z0, mark(z1)) → c107(U52'(z0, z1))
U52'(active(z0), z1) → c108(U52'(z0, z1))
U52'(z0, active(z1)) → c109(U52'(z0, z1))
U61'(mark(z0), z1, z2) → c110(U61'(z0, z1, z2))
U61'(z0, mark(z1), z2) → c111(U61'(z0, z1, z2))
U61'(z0, z1, mark(z2)) → c112(U61'(z0, z1, z2))
U61'(active(z0), z1, z2) → c113(U61'(z0, z1, z2))
U61'(z0, active(z1), z2) → c114(U61'(z0, z1, z2))
U61'(z0, z1, active(z2)) → c115(U61'(z0, z1, z2))
U62'(mark(z0), z1, z2) → c116(U62'(z0, z1, z2))
U62'(z0, mark(z1), z2) → c117(U62'(z0, z1, z2))
U62'(z0, z1, mark(z2)) → c118(U62'(z0, z1, z2))
U62'(active(z0), z1, z2) → c119(U62'(z0, z1, z2))
U62'(z0, active(z1), z2) → c120(U62'(z0, z1, z2))
U62'(z0, z1, active(z2)) → c121(U62'(z0, z1, z2))
U63'(mark(z0), z1, z2) → c122(U63'(z0, z1, z2))
U63'(z0, mark(z1), z2) → c123(U63'(z0, z1, z2))
U63'(z0, z1, mark(z2)) → c124(U63'(z0, z1, z2))
U63'(active(z0), z1, z2) → c125(U63'(z0, z1, z2))
U63'(z0, active(z1), z2) → c126(U63'(z0, z1, z2))
U63'(z0, z1, active(z2)) → c127(U63'(z0, z1, z2))
U64'(mark(z0), z1, z2) → c128(U64'(z0, z1, z2))
U64'(z0, mark(z1), z2) → c129(U64'(z0, z1, z2))
U64'(z0, z1, mark(z2)) → c130(U64'(z0, z1, z2))
U64'(active(z0), z1, z2) → c131(U64'(z0, z1, z2))
U64'(z0, active(z1), z2) → c132(U64'(z0, z1, z2))
U64'(z0, z1, active(z2)) → c133(U64'(z0, z1, z2))
S(mark(z0)) → c134(S(z0))
S(active(z0)) → c135(S(z0))
PLUS(mark(z0), z1) → c136(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c137(PLUS(z0, z1))
PLUS(active(z0), z1) → c138(PLUS(z0, z1))
PLUS(z0, active(z1)) → c139(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVE, MARK, U11', U12', ISNATKIND, U13', U14', U15', ISNAT, U16', U21', U22', U23', U31', U32', U41', U51', U52', U61', U62', U63', U64', S, PLUS

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110, c111, c112, c113, c114, c115, c116, c117, c118, c119, c120, c121, c122, c123, c124, c125, c126, c127, c128, c129, c130, c131, c132, c133, c134, c135, c136, c137, c138, c139

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(isNatKind(z0), z0, z1)), U12'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(U12(tt, z0, z1)) → c1(MARK(U13(isNatKind(z1), z0, z1)), U13'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U13(tt, z0, z1)) → c2(MARK(U14(isNatKind(z1), z0, z1)), U14'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U14(tt, z0, z1)) → c3(MARK(U15(isNat(z0), z1)), U15'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U15(tt, z0)) → c4(MARK(U16(isNat(z0))), U16'(isNat(z0)), ISNAT(z0))
ACTIVE(U16(tt)) → c5(MARK(tt))
ACTIVE(U21(tt, z0)) → c6(MARK(U22(isNatKind(z0), z0)), U22'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(U22(tt, z0)) → c7(MARK(U23(isNat(z0))), U23'(isNat(z0)), ISNAT(z0))
ACTIVE(U23(tt)) → c8(MARK(tt))
ACTIVE(U31(tt, z0)) → c9(MARK(U32(isNatKind(z0))), U32'(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(U32(tt)) → c10(MARK(tt))
ACTIVE(U41(tt)) → c11(MARK(tt))
ACTIVE(U51(tt, z0)) → c12(MARK(U52(isNatKind(z0), z0)), U52'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(U52(tt, z0)) → c13(MARK(z0))
ACTIVE(U61(tt, z0, z1)) → c14(MARK(U62(isNatKind(z0), z0, z1)), U62'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(U62(tt, z0, z1)) → c15(MARK(U63(isNat(z1), z0, z1)), U63'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U63(tt, z0, z1)) → c16(MARK(U64(isNatKind(z1), z0, z1)), U64'(isNatKind(z1), z0, z1), ISNATKIND(z1))
ACTIVE(U64(tt, z0, z1)) → c17(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(0)) → c18(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c19(MARK(U11(isNatKind(z0), z0, z1)), U11'(isNatKind(z0), z0, z1), ISNATKIND(z0))
ACTIVE(isNat(s(z0))) → c20(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c21(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c22(MARK(U31(isNatKind(z0), z1)), U31'(isNatKind(z0), z1), ISNATKIND(z0))
ACTIVE(isNatKind(s(z0))) → c23(MARK(U41(isNatKind(z0))), U41'(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c24(MARK(U51(isNat(z0), z0)), U51'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c25(MARK(U61(isNat(z1), z1, z0)), U61'(isNat(z1), z1, z0), ISNAT(z1))
MARK(U11(z0, z1, z2)) → c26(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(U12(z0, z1, z2)) → c28(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(isNatKind(z0)) → c29(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
MARK(U13(z0, z1, z2)) → c30(ACTIVE(U13(mark(z0), z1, z2)), U13'(mark(z0), z1, z2), MARK(z0))
MARK(U14(z0, z1, z2)) → c31(ACTIVE(U14(mark(z0), z1, z2)), U14'(mark(z0), z1, z2), MARK(z0))
MARK(U15(z0, z1)) → c32(ACTIVE(U15(mark(z0), z1)), U15'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c33(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U16(z0)) → c34(ACTIVE(U16(mark(z0))), U16'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c35(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0, z1)) → c36(ACTIVE(U22(mark(z0), z1)), U22'(mark(z0), z1), MARK(z0))
MARK(U23(z0)) → c37(ACTIVE(U23(mark(z0))), U23'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c38(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U32(z0)) → c39(ACTIVE(U32(mark(z0))), U32'(mark(z0)), MARK(z0))
MARK(U41(z0)) → c40(ACTIVE(U41(mark(z0))), U41'(mark(z0)), MARK(z0))
MARK(U51(z0, z1)) → c41(ACTIVE(U51(mark(z0), z1)), U51'(mark(z0), z1), MARK(z0))
MARK(U52(z0, z1)) → c42(ACTIVE(U52(mark(z0), z1)), U52'(mark(z0), z1), MARK(z0))
MARK(U61(z0, z1, z2)) → c43(ACTIVE(U61(mark(z0), z1, z2)), U61'(mark(z0), z1, z2), MARK(z0))
MARK(U62(z0, z1, z2)) → c44(ACTIVE(U62(mark(z0), z1, z2)), U62'(mark(z0), z1, z2), MARK(z0))
MARK(U63(z0, z1, z2)) → c45(ACTIVE(U63(mark(z0), z1, z2)), U63'(mark(z0), z1, z2), MARK(z0))
MARK(U64(z0, z1, z2)) → c46(ACTIVE(U64(mark(z0), z1, z2)), U64'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c47(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c48(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
U11'(mark(z0), z1, z2) → c50(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c51(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c52(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c53(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c54(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c55(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c56(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c57(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c58(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c59(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c60(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c61(U12'(z0, z1, z2))
ISNATKIND(mark(z0)) → c62(ISNATKIND(z0))
ISNATKIND(active(z0)) → c63(ISNATKIND(z0))
U13'(mark(z0), z1, z2) → c64(U13'(z0, z1, z2))
U13'(z0, mark(z1), z2) → c65(U13'(z0, z1, z2))
U13'(z0, z1, mark(z2)) → c66(U13'(z0, z1, z2))
U13'(active(z0), z1, z2) → c67(U13'(z0, z1, z2))
U13'(z0, active(z1), z2) → c68(U13'(z0, z1, z2))
U13'(z0, z1, active(z2)) → c69(U13'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c70(U14'(z0, z1, z2))
U14'(z0, mark(z1), z2) → c71(U14'(z0, z1, z2))
U14'(z0, z1, mark(z2)) → c72(U14'(z0, z1, z2))
U14'(active(z0), z1, z2) → c73(U14'(z0, z1, z2))
U14'(z0, active(z1), z2) → c74(U14'(z0, z1, z2))
U14'(z0, z1, active(z2)) → c75(U14'(z0, z1, z2))
U15'(mark(z0), z1) → c76(U15'(z0, z1))
U15'(z0, mark(z1)) → c77(U15'(z0, z1))
U15'(active(z0), z1) → c78(U15'(z0, z1))
U15'(z0, active(z1)) → c79(U15'(z0, z1))
ISNAT(mark(z0)) → c80(ISNAT(z0))
ISNAT(active(z0)) → c81(ISNAT(z0))
U16'(mark(z0)) → c82(U16'(z0))
U16'(active(z0)) → c83(U16'(z0))
U21'(mark(z0), z1) → c84(U21'(z0, z1))
U21'(z0, mark(z1)) → c85(U21'(z0, z1))
U21'(active(z0), z1) → c86(U21'(z0, z1))
U21'(z0, active(z1)) → c87(U21'(z0, z1))
U22'(mark(z0), z1) → c88(U22'(z0, z1))
U22'(z0, mark(z1)) → c89(U22'(z0, z1))
U22'(active(z0), z1) → c90(U22'(z0, z1))
U22'(z0, active(z1)) → c91(U22'(z0, z1))
U23'(mark(z0)) → c92(U23'(z0))
U23'(active(z0)) → c93(U23'(z0))
U31'(mark(z0), z1) → c94(U31'(z0, z1))
U31'(z0, mark(z1)) → c95(U31'(z0, z1))
U31'(active(z0), z1) → c96(U31'(z0, z1))
U31'(z0, active(z1)) → c97(U31'(z0, z1))
U32'(mark(z0)) → c98(U32'(z0))
U32'(active(z0)) → c99(U32'(z0))
U41'(mark(z0)) → c100(U41'(z0))
U41'(active(z0)) → c101(U41'(z0))
U51'(mark(z0), z1) → c102(U51'(z0, z1))
U51'(z0, mark(z1)) → c103(U51'(z0, z1))
U51'(active(z0), z1) → c104(U51'(z0, z1))
U51'(z0, active(z1)) → c105(U51'(z0, z1))
U52'(mark(z0), z1) → c106(U52'(z0, z1))
U52'(z0, mark(z1)) → c107(U52'(z0, z1))
U52'(active(z0), z1) → c108(U52'(z0, z1))
U52'(z0, active(z1)) → c109(U52'(z0, z1))
U61'(mark(z0), z1, z2) → c110(U61'(z0, z1, z2))
U61'(z0, mark(z1), z2) → c111(U61'(z0, z1, z2))
U61'(z0, z1, mark(z2)) → c112(U61'(z0, z1, z2))
U61'(active(z0), z1, z2) → c113(U61'(z0, z1, z2))
U61'(z0, active(z1), z2) → c114(U61'(z0, z1, z2))
U61'(z0, z1, active(z2)) → c115(U61'(z0, z1, z2))
U62'(mark(z0), z1, z2) → c116(U62'(z0, z1, z2))
U62'(z0, mark(z1), z2) → c117(U62'(z0, z1, z2))
U62'(z0, z1, mark(z2)) → c118(U62'(z0, z1, z2))
U62'(active(z0), z1, z2) → c119(U62'(z0, z1, z2))
U62'(z0, active(z1), z2) → c120(U62'(z0, z1, z2))
U62'(z0, z1, active(z2)) → c121(U62'(z0, z1, z2))
U63'(mark(z0), z1, z2) → c122(U63'(z0, z1, z2))
U63'(z0, mark(z1), z2) → c123(U63'(z0, z1, z2))
U63'(z0, z1, mark(z2)) → c124(U63'(z0, z1, z2))
U63'(active(z0), z1, z2) → c125(U63'(z0, z1, z2))
U63'(z0, active(z1), z2) → c126(U63'(z0, z1, z2))
U63'(z0, z1, active(z2)) → c127(U63'(z0, z1, z2))
U64'(mark(z0), z1, z2) → c128(U64'(z0, z1, z2))
U64'(z0, mark(z1), z2) → c129(U64'(z0, z1, z2))
U64'(z0, z1, mark(z2)) → c130(U64'(z0, z1, z2))
U64'(active(z0), z1, z2) → c131(U64'(z0, z1, z2))
U64'(z0, active(z1), z2) → c132(U64'(z0, z1, z2))
U64'(z0, z1, active(z2)) → c133(U64'(z0, z1, z2))
S(mark(z0)) → c134(S(z0))
S(active(z0)) → c135(S(z0))
PLUS(mark(z0), z1) → c136(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c137(PLUS(z0, z1))
PLUS(active(z0), z1) → c138(PLUS(z0, z1))
PLUS(z0, active(z1)) → c139(PLUS(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(isNatKind(z0), z0, z1))
active(U12(tt, z0, z1)) → mark(U13(isNatKind(z1), z0, z1))
active(U13(tt, z0, z1)) → mark(U14(isNatKind(z1), z0, z1))
active(U14(tt, z0, z1)) → mark(U15(isNat(z0), z1))
active(U15(tt, z0)) → mark(U16(isNat(z0)))
active(U16(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNatKind(z0), z0))
active(U22(tt, z0)) → mark(U23(isNat(z0)))
active(U23(tt)) → mark(tt)
active(U31(tt, z0)) → mark(U32(isNatKind(z0)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, z0)) → mark(U52(isNatKind(z0), z0))
active(U52(tt, z0)) → mark(z0)
active(U61(tt, z0, z1)) → mark(U62(isNatKind(z0), z0, z1))
active(U62(tt, z0, z1)) → mark(U63(isNat(z1), z0, z1))
active(U63(tt, z0, z1)) → mark(U64(isNatKind(z1), z0, z1))
active(U64(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNatKind(z0), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(U31(isNatKind(z0), z1))
active(isNatKind(s(z0))) → mark(U41(isNatKind(z0)))
active(plus(z0, 0)) → mark(U51(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U61(isNat(z1), z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1, z2)) → active(U12(mark(z0), z1, z2))
mark(isNatKind(z0)) → active(isNatKind(z0))
mark(U13(z0, z1, z2)) → active(U13(mark(z0), z1, z2))
mark(U14(z0, z1, z2)) → active(U14(mark(z0), z1, z2))
mark(U15(z0, z1)) → active(U15(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(U16(z0)) → active(U16(mark(z0)))
mark(U21(z0, z1)) → active(U21(mark(z0), z1))
mark(U22(z0, z1)) → active(U22(mark(z0), z1))
mark(U23(z0)) → active(U23(mark(z0)))
mark(U31(z0, z1)) → active(U31(mark(z0), z1))
mark(U32(z0)) → active(U32(mark(z0)))
mark(U41(z0)) → active(U41(mark(z0)))
mark(U51(z0, z1)) → active(U51(mark(z0), z1))
mark(U52(z0, z1)) → active(U52(mark(z0), z1))
mark(U61(z0, z1, z2)) → active(U61(mark(z0), z1, z2))
mark(U62(z0, z1, z2)) → active(U62(mark(z0), z1, z2))
mark(U63(z0, z1, z2)) → active(U63(mark(z0), z1, z2))
mark(U64(z0, z1, z2)) → active(U64(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(0) → active(0)
U11(mark(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, mark(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, mark(z2)) → U11(z0, z1, z2)
U11(active(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, active(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, active(z2)) → U11(z0, z1, z2)
U12(mark(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, mark(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, mark(z2)) → U12(z0, z1, z2)
U12(active(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, active(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, active(z2)) → U12(z0, z1, z2)
isNatKind(mark(z0)) → isNatKind(z0)
isNatKind(active(z0)) → isNatKind(z0)
U13(mark(z0), z1, z2) → U13(z0, z1, z2)
U13(z0, mark(z1), z2) → U13(z0, z1, z2)
U13(z0, z1, mark(z2)) → U13(z0, z1, z2)
U13(active(z0), z1, z2) → U13(z0, z1, z2)
U13(z0, active(z1), z2) → U13(z0, z1, z2)
U13(z0, z1, active(z2)) → U13(z0, z1, z2)
U14(mark(z0), z1, z2) → U14(z0, z1, z2)
U14(z0, mark(z1), z2) → U14(z0, z1, z2)
U14(z0, z1, mark(z2)) → U14(z0, z1, z2)
U14(active(z0), z1, z2) → U14(z0, z1, z2)
U14(z0, active(z1), z2) → U14(z0, z1, z2)
U14(z0, z1, active(z2)) → U14(z0, z1, z2)
U15(mark(z0), z1) → U15(z0, z1)
U15(z0, mark(z1)) → U15(z0, z1)
U15(active(z0), z1) → U15(z0, z1)
U15(z0, active(z1)) → U15(z0, z1)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U16(mark(z0)) → U16(z0)
U16(active(z0)) → U16(z0)
U21(mark(z0), z1) → U21(z0, z1)
U21(z0, mark(z1)) → U21(z0, z1)
U21(active(z0), z1) → U21(z0, z1)
U21(z0, active(z1)) → U21(z0, z1)
U22(mark(z0), z1) → U22(z0, z1)
U22(z0, mark(z1)) → U22(z0, z1)
U22(active(z0), z1) → U22(z0, z1)
U22(z0, active(z1)) → U22(z0, z1)
U23(mark(z0)) → U23(z0)
U23(active(z0)) → U23(z0)
U31(mark(z0), z1) → U31(z0, z1)
U31(z0, mark(z1)) → U31(z0, z1)
U31(active(z0), z1) → U31(z0, z1)
U31(z0, active(z1)) → U31(z0, z1)
U32(mark(z0)) → U32(z0)
U32(active(z0)) → U32(z0)
U41(mark(z0)) → U41(z0)
U41(active(z0)) → U41(z0)
U51(mark(z0), z1) → U51(z0, z1)
U51(z0, mark(z1)) → U51(z0, z1)
U51(active(z0), z1) → U51(z0, z1)
U51(z0, active(z1)) → U51(z0, z1)
U52(mark(z0), z1) → U52(z0, z1)
U52(z0, mark(z1)) → U52(z0, z1)
U52(active(z0), z1) → U52(z0, z1)
U52(z0, active(z1)) → U52(z0, z1)
U61(mark(z0), z1, z2) → U61(z0, z1, z2)
U61(z0, mark(z1), z2) → U61(z0, z1, z2)
U61(z0, z1, mark(z2)) → U61(z0, z1, z2)
U61(active(z0), z1, z2) → U61(z0, z1, z2)
U61(z0, active(z1), z2) → U61(z0, z1, z2)
U61(z0, z1, active(z2)) → U61(z0, z1, z2)
U62(mark(z0), z1, z2) → U62(z0, z1, z2)
U62(z0, mark(z1), z2) → U62(z0, z1, z2)
U62(z0, z1, mark(z2)) → U62(z0, z1, z2)
U62(active(z0), z1, z2) → U62(z0, z1, z2)
U62(z0, active(z1), z2) → U62(z0, z1, z2)
U62(z0, z1, active(z2)) → U62(z0, z1, z2)
U63(mark(z0), z1, z2) → U63(z0, z1, z2)
U63(z0, mark(z1), z2) → U63(z0, z1, z2)
U63(z0, z1, mark(z2)) → U63(z0, z1, z2)
U63(active(z0), z1, z2) → U63(z0, z1, z2)
U63(z0, active(z1), z2) → U63(z0, z1, z2)
U63(z0, z1, active(z2)) → U63(z0, z1, z2)
U64(mark(z0), z1, z2) → U64(z0, z1, z2)
U64(z0, mark(z1), z2) → U64(z0, z1, z2)
U64(z0, z1, mark(z2)) → U64(z0, z1, z2)
U64(active(z0), z1, z2) → U64(z0, z1, z2)
U64(z0, active(z1), z2) → U64(z0, z1, z2)
U64(z0, z1, active(z2)) → U64(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(z0, z1)

Tuples:

MARK(tt) → c27(ACTIVE(tt))
MARK(0) → c49(ACTIVE(0))

S tuples:

MARK(tt) → c27(ACTIVE(tt))
MARK(0) → c49(ACTIVE(0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

MARK

Compound Symbols:

c27, c49

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

MARK(0) → c49(ACTIVE(0))
MARK(tt) → c27(ACTIVE(tt))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(isNatKind(z0), z0, z1))
active(U12(tt, z0, z1)) → mark(U13(isNatKind(z1), z0, z1))
active(U13(tt, z0, z1)) → mark(U14(isNatKind(z1), z0, z1))
active(U14(tt, z0, z1)) → mark(U15(isNat(z0), z1))
active(U15(tt, z0)) → mark(U16(isNat(z0)))
active(U16(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNatKind(z0), z0))
active(U22(tt, z0)) → mark(U23(isNat(z0)))
active(U23(tt)) → mark(tt)
active(U31(tt, z0)) → mark(U32(isNatKind(z0)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, z0)) → mark(U52(isNatKind(z0), z0))
active(U52(tt, z0)) → mark(z0)
active(U61(tt, z0, z1)) → mark(U62(isNatKind(z0), z0, z1))
active(U62(tt, z0, z1)) → mark(U63(isNat(z1), z0, z1))
active(U63(tt, z0, z1)) → mark(U64(isNatKind(z1), z0, z1))
active(U64(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNatKind(z0), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(U31(isNatKind(z0), z1))
active(isNatKind(s(z0))) → mark(U41(isNatKind(z0)))
active(plus(z0, 0)) → mark(U51(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U61(isNat(z1), z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1, z2)) → active(U12(mark(z0), z1, z2))
mark(isNatKind(z0)) → active(isNatKind(z0))
mark(U13(z0, z1, z2)) → active(U13(mark(z0), z1, z2))
mark(U14(z0, z1, z2)) → active(U14(mark(z0), z1, z2))
mark(U15(z0, z1)) → active(U15(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(U16(z0)) → active(U16(mark(z0)))
mark(U21(z0, z1)) → active(U21(mark(z0), z1))
mark(U22(z0, z1)) → active(U22(mark(z0), z1))
mark(U23(z0)) → active(U23(mark(z0)))
mark(U31(z0, z1)) → active(U31(mark(z0), z1))
mark(U32(z0)) → active(U32(mark(z0)))
mark(U41(z0)) → active(U41(mark(z0)))
mark(U51(z0, z1)) → active(U51(mark(z0), z1))
mark(U52(z0, z1)) → active(U52(mark(z0), z1))
mark(U61(z0, z1, z2)) → active(U61(mark(z0), z1, z2))
mark(U62(z0, z1, z2)) → active(U62(mark(z0), z1, z2))
mark(U63(z0, z1, z2)) → active(U63(mark(z0), z1, z2))
mark(U64(z0, z1, z2)) → active(U64(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(0) → active(0)
U11(mark(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, mark(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, mark(z2)) → U11(z0, z1, z2)
U11(active(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, active(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, active(z2)) → U11(z0, z1, z2)
U12(mark(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, mark(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, mark(z2)) → U12(z0, z1, z2)
U12(active(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, active(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, active(z2)) → U12(z0, z1, z2)
isNatKind(mark(z0)) → isNatKind(z0)
isNatKind(active(z0)) → isNatKind(z0)
U13(mark(z0), z1, z2) → U13(z0, z1, z2)
U13(z0, mark(z1), z2) → U13(z0, z1, z2)
U13(z0, z1, mark(z2)) → U13(z0, z1, z2)
U13(active(z0), z1, z2) → U13(z0, z1, z2)
U13(z0, active(z1), z2) → U13(z0, z1, z2)
U13(z0, z1, active(z2)) → U13(z0, z1, z2)
U14(mark(z0), z1, z2) → U14(z0, z1, z2)
U14(z0, mark(z1), z2) → U14(z0, z1, z2)
U14(z0, z1, mark(z2)) → U14(z0, z1, z2)
U14(active(z0), z1, z2) → U14(z0, z1, z2)
U14(z0, active(z1), z2) → U14(z0, z1, z2)
U14(z0, z1, active(z2)) → U14(z0, z1, z2)
U15(mark(z0), z1) → U15(z0, z1)
U15(z0, mark(z1)) → U15(z0, z1)
U15(active(z0), z1) → U15(z0, z1)
U15(z0, active(z1)) → U15(z0, z1)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U16(mark(z0)) → U16(z0)
U16(active(z0)) → U16(z0)
U21(mark(z0), z1) → U21(z0, z1)
U21(z0, mark(z1)) → U21(z0, z1)
U21(active(z0), z1) → U21(z0, z1)
U21(z0, active(z1)) → U21(z0, z1)
U22(mark(z0), z1) → U22(z0, z1)
U22(z0, mark(z1)) → U22(z0, z1)
U22(active(z0), z1) → U22(z0, z1)
U22(z0, active(z1)) → U22(z0, z1)
U23(mark(z0)) → U23(z0)
U23(active(z0)) → U23(z0)
U31(mark(z0), z1) → U31(z0, z1)
U31(z0, mark(z1)) → U31(z0, z1)
U31(active(z0), z1) → U31(z0, z1)
U31(z0, active(z1)) → U31(z0, z1)
U32(mark(z0)) → U32(z0)
U32(active(z0)) → U32(z0)
U41(mark(z0)) → U41(z0)
U41(active(z0)) → U41(z0)
U51(mark(z0), z1) → U51(z0, z1)
U51(z0, mark(z1)) → U51(z0, z1)
U51(active(z0), z1) → U51(z0, z1)
U51(z0, active(z1)) → U51(z0, z1)
U52(mark(z0), z1) → U52(z0, z1)
U52(z0, mark(z1)) → U52(z0, z1)
U52(active(z0), z1) → U52(z0, z1)
U52(z0, active(z1)) → U52(z0, z1)
U61(mark(z0), z1, z2) → U61(z0, z1, z2)
U61(z0, mark(z1), z2) → U61(z0, z1, z2)
U61(z0, z1, mark(z2)) → U61(z0, z1, z2)
U61(active(z0), z1, z2) → U61(z0, z1, z2)
U61(z0, active(z1), z2) → U61(z0, z1, z2)
U61(z0, z1, active(z2)) → U61(z0, z1, z2)
U62(mark(z0), z1, z2) → U62(z0, z1, z2)
U62(z0, mark(z1), z2) → U62(z0, z1, z2)
U62(z0, z1, mark(z2)) → U62(z0, z1, z2)
U62(active(z0), z1, z2) → U62(z0, z1, z2)
U62(z0, active(z1), z2) → U62(z0, z1, z2)
U62(z0, z1, active(z2)) → U62(z0, z1, z2)
U63(mark(z0), z1, z2) → U63(z0, z1, z2)
U63(z0, mark(z1), z2) → U63(z0, z1, z2)
U63(z0, z1, mark(z2)) → U63(z0, z1, z2)
U63(active(z0), z1, z2) → U63(z0, z1, z2)
U63(z0, active(z1), z2) → U63(z0, z1, z2)
U63(z0, z1, active(z2)) → U63(z0, z1, z2)
U64(mark(z0), z1, z2) → U64(z0, z1, z2)
U64(z0, mark(z1), z2) → U64(z0, z1, z2)
U64(z0, z1, mark(z2)) → U64(z0, z1, z2)
U64(active(z0), z1, z2) → U64(z0, z1, z2)
U64(z0, active(z1), z2) → U64(z0, z1, z2)
U64(z0, z1, active(z2)) → U64(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(z0, z1)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))