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

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 40 SCCs with 144 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(ok(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

ISNAT₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(7) Obligation:

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

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

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNATKIND(ok(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

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

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(ok(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

ISNATKIND₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

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

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

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

U92¹(ok(X)) → U92¹(X)
U92¹(mark(X)) → U92¹(X)

The TRS R consists of the following rules:

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

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(mark(X)) → U92¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(17) Obligation:

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

U92¹(ok(X)) → U92¹(X)

The TRS R consists of the following rules:

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

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(ok(X)) → U92¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1)  =  U92¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U92^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(19) Obligation:

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

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

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

(20) PisEmptyProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(mark(X1), X2) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U12₃ > isNatKind > tt > U32₃ > mark₁
active₁ > U12₃ > isNatKind > tt > U33₃ > mark₁
active₁ > U13₃ > isNatKind > tt > U32₃ > mark₁
active₁ > U13₃ > isNatKind > tt > U33₃ > mark₁
active₁ > U14₃ > isNat₁ > isNatKind > tt > U32₃ > mark₁
active₁ > U14₃ > isNat₁ > isNatKind > tt > U33₃ > mark₁
active₁ > U14₃ > isNat₁ > U11₃ > mark₁
active₁ > U14₃ > isNat₁ > U21₂ > U22₂ > mark₁
active₁ > U14₃ > U15₂ > mark₁
active₁ > U23₁ > tt > U32₃ > mark₁
active₁ > U23₁ > tt > U33₃ > mark₁
active₁ > U31₃ > isNatKind > tt > U32₃ > mark₁
active₁ > U31₃ > isNatKind > tt > U33₃ > mark₁
active₁ > U34₃ > mark₁
active₁ > U35₂ > U36₁ > tt > U32₃ > mark₁
active₁ > U35₂ > U36₁ > tt > U33₃ > mark₁
active₁ > U51₁ > tt > U32₃ > mark₁
active₁ > U51₁ > tt > U33₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U32₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U33₃ > mark₁
active₁ > U71₂ > U72₂ > mark₁
active₁ > U83₃ > U84₃ > plus₂ > U11₃ > mark₁
active₁ > U83₃ > U84₃ > plus₂ > U81₃ > U82₃ > mark₁
active₁ > s₁ > U101₃ > U102₃ > mark₁
active₁ > s₁ > U21₂ > U22₂ > mark₁
active₁ > s₁ > U81₃ > U82₃ > mark₁
active₁ > 0 > mark₁
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U12₃ > isNatKind > tt > U32₃ > mark₁
proper₁ > U12₃ > isNatKind > tt > U33₃ > mark₁
proper₁ > U13₃ > isNatKind > tt > U32₃ > mark₁
proper₁ > U13₃ > isNatKind > tt > U33₃ > mark₁
proper₁ > U14₃ > isNat₁ > isNatKind > tt > U32₃ > mark₁
proper₁ > U14₃ > isNat₁ > isNatKind > tt > U33₃ > mark₁
proper₁ > U14₃ > isNat₁ > U11₃ > mark₁
proper₁ > U14₃ > isNat₁ > U21₂ > U22₂ > mark₁
proper₁ > U14₃ > U15₂ > mark₁
proper₁ > U23₁ > tt > U32₃ > mark₁
proper₁ > U23₁ > tt > U33₃ > mark₁
proper₁ > U31₃ > isNatKind > tt > U32₃ > mark₁
proper₁ > U31₃ > isNatKind > tt > U33₃ > mark₁
proper₁ > U34₃ > mark₁
proper₁ > U35₂ > U36₁ > tt > U32₃ > mark₁
proper₁ > U35₂ > U36₁ > tt > U33₃ > mark₁
proper₁ > U51₁ > tt > U32₃ > mark₁
proper₁ > U51₁ > tt > U33₃ > mark₁
proper₁ > U71₂ > isNatKind > tt > U32₃ > mark₁
proper₁ > U71₂ > isNatKind > tt > U33₃ > mark₁
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U83₃ > U84₃ > plus₂ > U11₃ > mark₁
proper₁ > U83₃ > U84₃ > plus₂ > U81₃ > U82₃ > mark₁
proper₁ > s₁ > U101₃ > U102₃ > mark₁
proper₁ > s₁ > U21₂ > U22₂ > mark₁
proper₁ > s₁ > U81₃ > U82₃ > mark₁
proper₁ > 0 > mark₁

The following usable rules [FROCOS05] were oriented:

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

(24) Obligation:

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

U91¹(ok(X1), ok(X2)) → U91¹(X1, X2)

The TRS R consists of the following rules:

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

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(ok(X1), ok(X2)) → U91¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2)  =  U91¹(x2)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U101(x1, x2, x3)  =  U101(x1, x2)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  x2
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  x2
x(x1, x2)  =  x2
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1, x2)
U15(x1, x2)  =  U15(x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x3)
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  x2
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  U35(x1)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1)
U72(x1, x2)  =  U72(x1)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U91^1₁ > mark
proper₁ > isNatKind₁ > tt > U103₂ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U104₃ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U12₃ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U14₂ > U15₁ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U34₁ > U35₁ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U72₁ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U82₁ > ok₁ > mark
proper₁ > isNatKind₁ > tt > U83₃ > ok₁ > mark
proper₁ > isNatKind₁ > tt > 0 > U71₁ > ok₁ > mark
proper₁ > isNatKind₁ > U41₁ > ok₁ > mark
proper₁ > isNatKind₁ > U61₂ > ok₁ > mark
proper₁ > U13₁ > U14₂ > U15₁ > ok₁ > mark
proper₁ > U23₁ > tt > U103₂ > ok₁ > mark
proper₁ > U23₁ > tt > U104₃ > ok₁ > mark
proper₁ > U23₁ > tt > U12₃ > ok₁ > mark
proper₁ > U23₁ > tt > U14₂ > U15₁ > ok₁ > mark
proper₁ > U23₁ > tt > U34₁ > U35₁ > ok₁ > mark
proper₁ > U23₁ > tt > U72₁ > ok₁ > mark
proper₁ > U23₁ > tt > U82₁ > ok₁ > mark
proper₁ > U23₁ > tt > U83₃ > ok₁ > mark
proper₁ > U23₁ > tt > 0 > U71₁ > ok₁ > mark
proper₁ > U31₂ > ok₁ > mark
proper₁ > U42₁ > tt > U103₂ > ok₁ > mark
proper₁ > U42₁ > tt > U104₃ > ok₁ > mark
proper₁ > U42₁ > tt > U12₃ > ok₁ > mark
proper₁ > U42₁ > tt > U14₂ > U15₁ > ok₁ > mark
proper₁ > U42₁ > tt > U34₁ > U35₁ > ok₁ > mark
proper₁ > U42₁ > tt > U72₁ > ok₁ > mark
proper₁ > U42₁ > tt > U82₁ > ok₁ > mark
proper₁ > U42₁ > tt > U83₃ > ok₁ > mark
proper₁ > U42₁ > tt > 0 > U71₁ > ok₁ > mark
proper₁ > U84₁ > s₁ > U101₂ > ok₁ > mark
proper₁ > U84₁ > s₁ > U21₂ > ok₁ > mark
proper₁ > U84₁ > s₁ > U81₃ > U82₁ > ok₁ > mark
top > mark

The following usable rules [FROCOS05] were oriented:

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

(26) Obligation:

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

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

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

(27) PisEmptyProof (EQUIVALENT transformation)

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

(28) TRUE

(29) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(31) Obligation:

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

S(ok(X)) → S(X)

The TRS R consists of the following rules:

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

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

S₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(33) Obligation:

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

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

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

(34) PisEmptyProof (EQUIVALENT transformation)

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

(35) TRUE

(36) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > U102₃ > isNat > tt > plus₂ > mark₁
top > active₁ > U102₃ > isNat > U31₁ > mark₁
top > active₁ > isNatKind > tt > plus₂ > mark₁
top > active₁ > U103₃ > U104₃ > plus₂ > mark₁
top > active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > active₁ > U103₃ > U104₃ > x₂ > U91₂ > mark₁
top > active₁ > U13₁ > mark₁
top > active₁ > U33₁ > mark₁
top > active₁ > U81₃ > U82₃ > mark₁
top > active₁ > U83₃ > U84₃ > plus₂ > mark₁
top > active₁ > U83₃ > U84₃ > s₁ > U101₃ > mark₁
top > active₁ > U92₁ > 0 > tt > plus₂ > mark₁
top > active₁ > U92₁ > 0 > U71₂ > U72₂ > mark₁
top > active₁ > U92₁ > 0 > U91₂ > mark₁
top > proper₁ > U102₃ > isNat > tt > plus₂ > mark₁
top > proper₁ > U102₃ > isNat > U31₁ > mark₁
top > proper₁ > isNatKind > tt > plus₂ > mark₁
top > proper₁ > U103₃ > U104₃ > plus₂ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U91₂ > mark₁
top > proper₁ > U13₁ > mark₁
top > proper₁ > U33₁ > mark₁
top > proper₁ > U81₃ > U82₃ > mark₁
top > proper₁ > U83₃ > U84₃ > plus₂ > mark₁
top > proper₁ > U83₃ > U84₃ > s₁ > U101₃ > mark₁
top > proper₁ > U92₁ > 0 > tt > plus₂ > mark₁
top > proper₁ > U92₁ > 0 > U71₂ > U72₂ > mark₁
top > proper₁ > U92₁ > 0 > U91₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(38) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x2, x3)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U102₃ > U103₃ > ok₁
active₁ > U102₃ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U11₃ > ok₁
active₁ > U104₃ > plus₂ > U11₃ > mark₁
active₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U35₂ > ok₁
active₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U35₂ > mark₁
active₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U84₃ > ok₁
active₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U84₃ > mark₁
active₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > U41₂ > ok₁
active₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > U41₂ > mark₁
active₁ > U104₃ > plus₂ > U71₂ > U72₂ > ok₁
active₁ > U104₃ > plus₂ > U71₂ > U72₂ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > ok₁
active₁ > U104₃ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U104₃ > x₂ > U91₂ > ok₁
active₁ > U104₃ > x₂ > U91₂ > mark₁
active₁ > U12₃ > U13₃ > ok₁
active₁ > U12₃ > U13₃ > mark₁
active₁ > U14₃ > ok₁
active₁ > U14₃ > mark₁
active₁ > U15₂ > ok₁
active₁ > U15₂ > mark₁
active₁ > U22₂ > U23₁ > tt > U35₂ > ok₁
active₁ > U22₂ > U23₁ > tt > U35₂ > mark₁
active₁ > U22₂ > U23₁ > tt > U84₃ > ok₁
active₁ > U22₂ > U23₁ > tt > U84₃ > mark₁
active₁ > U31₃ > U32₃ > isNatKind₁ > tt > U35₂ > ok₁
active₁ > U31₃ > U32₃ > isNatKind₁ > tt > U35₂ > mark₁
active₁ > U31₃ > U32₃ > isNatKind₁ > tt > U84₃ > ok₁
active₁ > U31₃ > U32₃ > isNatKind₁ > tt > U84₃ > mark₁
active₁ > U31₃ > U32₃ > isNatKind₁ > U41₂ > ok₁
active₁ > U31₃ > U32₃ > isNatKind₁ > U41₂ > mark₁
active₁ > U31₃ > U32₃ > U33₃ > ok₁
active₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U34₃ > U35₂ > ok₁
active₁ > U34₃ > U35₂ > mark₁
active₁ > U42₁ > tt > U35₂ > ok₁
active₁ > U42₁ > tt > U35₂ > mark₁
active₁ > U42₁ > tt > U84₃ > ok₁
active₁ > U42₁ > tt > U84₃ > mark₁
active₁ > U51₁ > tt > U35₂ > ok₁
active₁ > U51₁ > tt > U35₂ > mark₁
active₁ > U51₁ > tt > U84₃ > ok₁
active₁ > U51₁ > tt > U84₃ > mark₁
active₁ > U61₂ > isNatKind₁ > tt > U35₂ > ok₁
active₁ > U61₂ > isNatKind₁ > tt > U35₂ > mark₁
active₁ > U61₂ > isNatKind₁ > tt > U84₃ > ok₁
active₁ > U61₂ > isNatKind₁ > tt > U84₃ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > ok₁
active₁ > U61₂ > isNatKind₁ > U41₂ > mark₁
active₁ > s₁ > U101₃ > isNatKind₁ > tt > U35₂ > ok₁
active₁ > s₁ > U101₃ > isNatKind₁ > tt > U35₂ > mark₁
active₁ > s₁ > U101₃ > isNatKind₁ > tt > U84₃ > ok₁
active₁ > s₁ > U101₃ > isNatKind₁ > tt > U84₃ > mark₁
active₁ > s₁ > U101₃ > isNatKind₁ > U41₂ > ok₁
active₁ > s₁ > U101₃ > isNatKind₁ > U41₂ > mark₁
active₁ > s₁ > U21₂ > ok₁
active₁ > s₁ > U21₂ > mark₁
active₁ > s₁ > U81₃ > U82₃ > U83₃ > U84₃ > ok₁
active₁ > s₁ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > 0 > U71₂ > isNatKind₁ > tt > U35₂ > ok₁
active₁ > 0 > U71₂ > isNatKind₁ > tt > U35₂ > mark₁
active₁ > 0 > U71₂ > isNatKind₁ > tt > U84₃ > ok₁
active₁ > 0 > U71₂ > isNatKind₁ > tt > U84₃ > mark₁
active₁ > 0 > U71₂ > isNatKind₁ > U41₂ > ok₁
active₁ > 0 > U71₂ > isNatKind₁ > U41₂ > mark₁
active₁ > 0 > U71₂ > U72₂ > ok₁
active₁ > 0 > U71₂ > U72₂ > mark₁
active₁ > 0 > U91₂ > ok₁
active₁ > 0 > U91₂ > mark₁
proper₁ > U102₃ > U103₃ > ok₁
proper₁ > U102₃ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U11₃ > ok₁
proper₁ > U104₃ > plus₂ > U11₃ > mark₁
proper₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U35₂ > ok₁
proper₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U35₂ > mark₁
proper₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U84₃ > ok₁
proper₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > tt > U84₃ > mark₁
proper₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > U41₂ > ok₁
proper₁ > U104₃ > plus₂ > U71₂ > isNatKind₁ > U41₂ > mark₁
proper₁ > U104₃ > plus₂ > U71₂ > U72₂ > ok₁
proper₁ > U104₃ > plus₂ > U71₂ > U72₂ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > ok₁
proper₁ > U104₃ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
proper₁ > U104₃ > x₂ > U91₂ > ok₁
proper₁ > U104₃ > x₂ > U91₂ > mark₁
proper₁ > U12₃ > U13₃ > ok₁
proper₁ > U12₃ > U13₃ > mark₁
proper₁ > U14₃ > ok₁
proper₁ > U14₃ > mark₁
proper₁ > U15₂ > ok₁
proper₁ > U15₂ > mark₁
proper₁ > U22₂ > U23₁ > tt > U35₂ > ok₁
proper₁ > U22₂ > U23₁ > tt > U35₂ > mark₁
proper₁ > U22₂ > U23₁ > tt > U84₃ > ok₁
proper₁ > U22₂ > U23₁ > tt > U84₃ > mark₁
proper₁ > U31₃ > U32₃ > isNatKind₁ > tt > U35₂ > ok₁
proper₁ > U31₃ > U32₃ > isNatKind₁ > tt > U35₂ > mark₁
proper₁ > U31₃ > U32₃ > isNatKind₁ > tt > U84₃ > ok₁
proper₁ > U31₃ > U32₃ > isNatKind₁ > tt > U84₃ > mark₁
proper₁ > U31₃ > U32₃ > isNatKind₁ > U41₂ > ok₁
proper₁ > U31₃ > U32₃ > isNatKind₁ > U41₂ > mark₁
proper₁ > U31₃ > U32₃ > U33₃ > ok₁
proper₁ > U31₃ > U32₃ > U33₃ > mark₁
proper₁ > U34₃ > U35₂ > ok₁
proper₁ > U34₃ > U35₂ > mark₁
proper₁ > U42₁ > tt > U35₂ > ok₁
proper₁ > U42₁ > tt > U35₂ > mark₁
proper₁ > U42₁ > tt > U84₃ > ok₁
proper₁ > U42₁ > tt > U84₃ > mark₁
proper₁ > U51₁ > tt > U35₂ > ok₁
proper₁ > U51₁ > tt > U35₂ > mark₁
proper₁ > U51₁ > tt > U84₃ > ok₁
proper₁ > U51₁ > tt > U84₃ > mark₁
proper₁ > U61₂ > isNatKind₁ > tt > U35₂ > ok₁
proper₁ > U61₂ > isNatKind₁ > tt > U35₂ > mark₁
proper₁ > U61₂ > isNatKind₁ > tt > U84₃ > ok₁
proper₁ > U61₂ > isNatKind₁ > tt > U84₃ > mark₁
proper₁ > U61₂ > isNatKind₁ > U41₂ > ok₁
proper₁ > U61₂ > isNatKind₁ > U41₂ > mark₁
proper₁ > s₁ > U101₃ > isNatKind₁ > tt > U35₂ > ok₁
proper₁ > s₁ > U101₃ > isNatKind₁ > tt > U35₂ > mark₁
proper₁ > s₁ > U101₃ > isNatKind₁ > tt > U84₃ > ok₁
proper₁ > s₁ > U101₃ > isNatKind₁ > tt > U84₃ > mark₁
proper₁ > s₁ > U101₃ > isNatKind₁ > U41₂ > ok₁
proper₁ > s₁ > U101₃ > isNatKind₁ > U41₂ > mark₁
proper₁ > s₁ > U21₂ > ok₁
proper₁ > s₁ > U21₂ > mark₁
proper₁ > s₁ > U81₃ > U82₃ > U83₃ > U84₃ > ok₁
proper₁ > s₁ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
proper₁ > 0 > U71₂ > isNatKind₁ > tt > U35₂ > ok₁
proper₁ > 0 > U71₂ > isNatKind₁ > tt > U35₂ > mark₁
proper₁ > 0 > U71₂ > isNatKind₁ > tt > U84₃ > ok₁
proper₁ > 0 > U71₂ > isNatKind₁ > tt > U84₃ > mark₁
proper₁ > 0 > U71₂ > isNatKind₁ > U41₂ > ok₁
proper₁ > 0 > U71₂ > isNatKind₁ > U41₂ > mark₁
proper₁ > 0 > U71₂ > U72₂ > ok₁
proper₁ > 0 > U71₂ > U72₂ > mark₁
proper₁ > 0 > U91₂ > ok₁
proper₁ > 0 > U91₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(40) Obligation:

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

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

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

(41) PisEmptyProof (EQUIVALENT transformation)

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

(42) TRUE

(43) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U83^1₃ > mark₁
active₁ > U102₃ > U103₃ > isNatKind > tt > plus₂ > mark₁
active₁ > U102₃ > U103₃ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U102₃ > U103₃ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U102₃ > U103₃ > isNatKind > tt > U23₁ > mark₁
active₁ > U102₃ > U103₃ > isNatKind > tt > U82₃ > mark₁
active₁ > U102₃ > U103₃ > isNatKind > U41₁ > mark₁
active₁ > U102₃ > U103₃ > U104₃ > plus₂ > mark₁
active₁ > U102₃ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > U102₃ > U103₃ > U104₃ > x₂ > U61₁ > U62₁ > mark₁
active₁ > isNat₁ > U31₃ > isNatKind > tt > plus₂ > mark₁
active₁ > isNat₁ > U31₃ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > isNat₁ > U31₃ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > isNat₁ > U31₃ > isNatKind > tt > U23₁ > mark₁
active₁ > isNat₁ > U31₃ > isNatKind > tt > U82₃ > mark₁
active₁ > isNat₁ > U31₃ > isNatKind > U41₁ > mark₁
active₁ > U11₃ > U12₃ > isNatKind > tt > plus₂ > mark₁
active₁ > U11₃ > U12₃ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U11₃ > U12₃ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U11₃ > U12₃ > isNatKind > tt > U23₁ > mark₁
active₁ > U11₃ > U12₃ > isNatKind > tt > U82₃ > mark₁
active₁ > U11₃ > U12₃ > isNatKind > U41₁ > mark₁
active₁ > U13₃ > isNatKind > tt > plus₂ > mark₁
active₁ > U13₃ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U13₃ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U13₃ > isNatKind > tt > U23₁ > mark₁
active₁ > U13₃ > isNatKind > tt > U82₃ > mark₁
active₁ > U13₃ > isNatKind > U41₁ > mark₁
active₁ > U13₃ > U14₃ > mark₁
active₁ > U15₂ > mark₁
active₁ > U21₂ > isNatKind > tt > plus₂ > mark₁
active₁ > U21₂ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U21₂ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U21₂ > isNatKind > tt > U23₁ > mark₁
active₁ > U21₂ > isNatKind > tt > U82₃ > mark₁
active₁ > U21₂ > isNatKind > U41₁ > mark₁
active₁ > U21₂ > U22₂ > U23₁ > mark₁
active₁ > U32₃ > isNatKind > tt > plus₂ > mark₁
active₁ > U32₃ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U32₃ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U32₃ > isNatKind > tt > U23₁ > mark₁
active₁ > U32₃ > isNatKind > tt > U82₃ > mark₁
active₁ > U32₃ > isNatKind > U41₁ > mark₁
active₁ > U32₃ > U33₃ > mark₁
active₁ > U34₃ > U35₂ > mark₁
active₁ > U71₂ > isNatKind > tt > plus₂ > mark₁
active₁ > U71₂ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U71₂ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U23₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U82₃ > mark₁
active₁ > U71₂ > isNatKind > U41₁ > mark₁
active₁ > U71₂ > U72₂ > mark₁
active₁ > U81₃ > isNatKind > tt > plus₂ > mark₁
active₁ > U81₃ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > U81₃ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > U81₃ > isNatKind > tt > U23₁ > mark₁
active₁ > U81₃ > isNatKind > tt > U82₃ > mark₁
active₁ > U81₃ > isNatKind > U41₁ > mark₁
active₁ > U83₃ > mark₁
active₁ > U84₃ > plus₂ > mark₁
active₁ > 0 > U91₂ > isNatKind > tt > plus₂ > mark₁
active₁ > 0 > U91₂ > isNatKind > tt > x₂ > U101₃ > mark₁
active₁ > 0 > U91₂ > isNatKind > tt > x₂ > U61₁ > U62₁ > mark₁
active₁ > 0 > U91₂ > isNatKind > tt > U23₁ > mark₁
active₁ > 0 > U91₂ > isNatKind > tt > U82₃ > mark₁
active₁ > 0 > U91₂ > isNatKind > U41₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(45) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x1, x3)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > ok₁ > U83^1₂
active₁ > U101₃ > mark₁ > U83^1₂
active₁ > U102₃ > U103₃ > U104₃ > ok₁ > U83^1₂
active₁ > U102₃ > U103₃ > U104₃ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > U11₃ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > U11₃ > mark₁ > U83^1₂
active₁ > U102₃ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
active₁ > U102₃ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > 0 > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > tt > 0 > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > U11₃ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > U11₃ > mark₁ > U83^1₂
active₁ > plus₂ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
active₁ > plus₂ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
active₁ > plus₂ > U71₂ > U72₂ > ok₁ > U83^1₂
active₁ > plus₂ > U71₂ > U72₂ > mark₁ > U83^1₂
active₁ > plus₂ > U81₃ > ok₁ > U83^1₂
active₁ > plus₂ > U81₃ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > 0 > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > tt > 0 > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > U11₃ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > U11₃ > mark₁ > U83^1₂
active₁ > x₂ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
active₁ > x₂ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
active₁ > x₂ > U61₂ > ok₁ > U83^1₂
active₁ > x₂ > U61₂ > mark₁ > U83^1₂
active₁ > U12₃ > U13₃ > ok₁ > U83^1₂
active₁ > U12₃ > U13₃ > mark₁ > U83^1₂
active₁ > U14₃ > U15₂ > ok₁ > U83^1₂
active₁ > U14₃ > U15₂ > mark₁ > U83^1₂
active₁ > U16₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U16₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U16₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U16₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U16₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U16₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U16₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U16₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U16₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U16₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U16₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U16₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U16₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U16₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U23₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U23₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U23₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U23₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U23₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U23₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U23₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U23₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U23₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U23₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U23₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U23₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U23₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U23₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U31₃ > U32₃ > ok₁ > U83^1₂
active₁ > U31₃ > U32₃ > mark₁ > U83^1₂
active₁ > U33₃ > ok₁ > U83^1₂
active₁ > U33₃ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > U11₃ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > U11₃ > mark₁ > U83^1₂
active₁ > U34₃ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
active₁ > U34₃ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
active₁ > U34₃ > U35₂ > ok₁ > U83^1₂
active₁ > U34₃ > U35₂ > mark₁ > U83^1₂
active₁ > U36₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U36₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U36₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U36₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U36₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U36₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U36₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U36₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U36₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U36₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U36₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U36₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U36₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U36₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U41₂ > ok₁ > U83^1₂
active₁ > U41₂ > mark₁ > U83^1₂
active₁ > U51₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U51₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U51₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U51₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U51₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U51₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U51₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U51₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U51₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U51₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U51₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U51₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U51₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U51₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U62₁ > tt > U104₃ > ok₁ > U83^1₂
active₁ > U62₁ > tt > U104₃ > mark₁ > U83^1₂
active₁ > U62₁ > tt > U13₃ > ok₁ > U83^1₂
active₁ > U62₁ > tt > U13₃ > mark₁ > U83^1₂
active₁ > U62₁ > tt > U15₂ > ok₁ > U83^1₂
active₁ > U62₁ > tt > U15₂ > mark₁ > U83^1₂
active₁ > U62₁ > tt > U22₂ > ok₁ > U83^1₂
active₁ > U62₁ > tt > U22₂ > mark₁ > U83^1₂
active₁ > U62₁ > tt > U83₃ > ok₁ > U83^1₂
active₁ > U62₁ > tt > U83₃ > mark₁ > U83^1₂
active₁ > U62₁ > tt > U84₃ > ok₁ > U83^1₂
active₁ > U62₁ > tt > U84₃ > mark₁ > U83^1₂
active₁ > U62₁ > tt > 0 > ok₁ > U83^1₂
active₁ > U62₁ > tt > 0 > mark₁ > U83^1₂
active₁ > U82₃ > U83₃ > ok₁ > U83^1₂
active₁ > U82₃ > U83₃ > mark₁ > U83^1₂
active₁ > U91₂ > ok₁ > U83^1₂
active₁ > U91₂ > mark₁ > U83^1₂
proper₁ > U101₃ > ok₁ > U83^1₂
proper₁ > U101₃ > mark₁ > U83^1₂
proper₁ > U102₃ > U103₃ > U104₃ > ok₁ > U83^1₂
proper₁ > U102₃ > U103₃ > U104₃ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > U11₃ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > U11₃ > mark₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
proper₁ > U102₃ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > U11₃ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > U11₃ > mark₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
proper₁ > plus₂ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
proper₁ > plus₂ > U71₂ > U72₂ > ok₁ > U83^1₂
proper₁ > plus₂ > U71₂ > U72₂ > mark₁ > U83^1₂
proper₁ > plus₂ > U81₃ > ok₁ > U83^1₂
proper₁ > plus₂ > U81₃ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > U11₃ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > U11₃ > mark₁ > U83^1₂
proper₁ > x₂ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
proper₁ > x₂ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
proper₁ > x₂ > U61₂ > ok₁ > U83^1₂
proper₁ > x₂ > U61₂ > mark₁ > U83^1₂
proper₁ > U12₃ > U13₃ > ok₁ > U83^1₂
proper₁ > U12₃ > U13₃ > mark₁ > U83^1₂
proper₁ > U14₃ > U15₂ > ok₁ > U83^1₂
proper₁ > U14₃ > U15₂ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U16₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U16₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U16₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U16₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U16₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U16₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U16₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U16₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U23₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U23₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U23₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U23₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U23₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U23₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U23₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U23₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U23₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U23₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U23₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U23₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U23₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U23₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U31₃ > U32₃ > ok₁ > U83^1₂
proper₁ > U31₃ > U32₃ > mark₁ > U83^1₂
proper₁ > U33₃ > ok₁ > U83^1₂
proper₁ > U33₃ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > U11₃ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > U11₃ > mark₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > U21₂ > U22₂ > ok₁ > U83^1₂
proper₁ > U34₃ > isNat₁ > U21₂ > U22₂ > mark₁ > U83^1₂
proper₁ > U34₃ > U35₂ > ok₁ > U83^1₂
proper₁ > U34₃ > U35₂ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U36₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U36₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U36₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U36₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U36₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U36₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U36₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U36₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U41₂ > ok₁ > U83^1₂
proper₁ > U41₂ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U51₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U51₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U51₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U51₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U51₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U51₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U51₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U51₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U62₁ > tt > U104₃ > ok₁ > U83^1₂
proper₁ > U62₁ > tt > U104₃ > mark₁ > U83^1₂
proper₁ > U62₁ > tt > U13₃ > ok₁ > U83^1₂
proper₁ > U62₁ > tt > U13₃ > mark₁ > U83^1₂
proper₁ > U62₁ > tt > U15₂ > ok₁ > U83^1₂
proper₁ > U62₁ > tt > U15₂ > mark₁ > U83^1₂
proper₁ > U62₁ > tt > U22₂ > ok₁ > U83^1₂
proper₁ > U62₁ > tt > U22₂ > mark₁ > U83^1₂
proper₁ > U62₁ > tt > U83₃ > ok₁ > U83^1₂
proper₁ > U62₁ > tt > U83₃ > mark₁ > U83^1₂
proper₁ > U62₁ > tt > U84₃ > ok₁ > U83^1₂
proper₁ > U62₁ > tt > U84₃ > mark₁ > U83^1₂
proper₁ > U62₁ > tt > 0 > ok₁ > U83^1₂
proper₁ > U62₁ > tt > 0 > mark₁ > U83^1₂
proper₁ > U82₃ > U83₃ > ok₁ > U83^1₂
proper₁ > U82₃ > U83₃ > mark₁ > U83^1₂
proper₁ > U91₂ > ok₁ > U83^1₂
proper₁ > U91₂ > mark₁ > U83^1₂
top > U83^1₂

The following usable rules [FROCOS05] were oriented:

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

(47) Obligation:

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

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

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

(48) PisEmptyProof (EQUIVALENT transformation)

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

(49) TRUE

(50) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U82^1₂ > mark₁
active₁ > U102₃ > mark₁
active₁ > U103₃ > U104₃ > plus₂ > U71₂ > mark₁
active₁ > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U31₃ > mark₁
active₁ > U11₃ > U12₃ > mark₁
active₁ > U13₃ > mark₁
active₁ > U14₃ > U15₂ > mark₁
active₁ > U22₂ > mark₁
active₁ > U33₃ > mark₁
active₁ > U34₃ > U35₂ > mark₁
active₁ > U41₂ > U42₁ > tt > U15₂ > mark₁
active₁ > U41₂ > U42₁ > tt > U32₃ > mark₁
active₁ > U41₂ > U42₁ > tt > U72₂ > mark₁
active₁ > U51₁ > tt > U15₂ > mark₁
active₁ > U51₁ > tt > U32₃ > mark₁
active₁ > U51₁ > tt > U72₂ > mark₁
active₁ > U61₂ > U62₁ > tt > U15₂ > mark₁
active₁ > U61₂ > U62₁ > tt > U32₃ > mark₁
active₁ > U61₂ > U62₁ > tt > U72₂ > mark₁
active₁ > U82₃ > mark₁
active₁ > U83₃ > U84₃ > mark₁
active₁ > s₁ > U81₃ > mark₁
active₁ > 0 > isNat₁ > tt > U15₂ > mark₁
active₁ > 0 > isNat₁ > tt > U32₃ > mark₁
active₁ > 0 > isNat₁ > tt > U72₂ > mark₁
active₁ > 0 > isNat₁ > U21₂ > mark₁
active₁ > 0 > isNat₁ > U31₃ > mark₁
active₁ > 0 > U71₂ > mark₁
active₁ > 0 > U91₂ > U92₁ > mark₁
proper₁ > U102₃ > mark₁
proper₁ > U103₃ > U104₃ > plus₂ > U71₂ > mark₁
proper₁ > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
proper₁ > U103₃ > U104₃ > x₂ > U31₃ > mark₁
proper₁ > isNat₁ > tt > U15₂ > mark₁
proper₁ > isNat₁ > tt > U32₃ > mark₁
proper₁ > isNat₁ > tt > U72₂ > mark₁
proper₁ > isNat₁ > U21₂ > mark₁
proper₁ > isNat₁ > U31₃ > mark₁
proper₁ > U11₃ > U12₃ > mark₁
proper₁ > U13₃ > mark₁
proper₁ > U14₃ > U15₂ > mark₁
proper₁ > U22₂ > mark₁
proper₁ > U33₃ > mark₁
proper₁ > U34₃ > U35₂ > mark₁
proper₁ > U41₂ > U42₁ > tt > U15₂ > mark₁
proper₁ > U41₂ > U42₁ > tt > U32₃ > mark₁
proper₁ > U41₂ > U42₁ > tt > U72₂ > mark₁
proper₁ > U51₁ > tt > U15₂ > mark₁
proper₁ > U51₁ > tt > U32₃ > mark₁
proper₁ > U51₁ > tt > U72₂ > mark₁
proper₁ > U61₂ > U62₁ > tt > U15₂ > mark₁
proper₁ > U61₂ > U62₁ > tt > U32₃ > mark₁
proper₁ > U61₂ > U62₁ > tt > U72₂ > mark₁
proper₁ > U82₃ > mark₁
proper₁ > U83₃ > U84₃ > mark₁
proper₁ > s₁ > U81₃ > mark₁
proper₁ > U91₂ > U92₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(52) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  x2
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  U102(x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2)
plus(x1, x2)  =  x2
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  U12(x1, x2)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x2, x3)
U15(x1, x2)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x2
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x2, x3)
U32(x1, x2, x3)  =  U32(x2, x3)
U33(x1, x2, x3)  =  x2
U34(x1, x2, x3)  =  U34(x2)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1)
U72(x1, x2)  =  U72(x1)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  x1
s(x1)  =  s(x1)
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > U102₁ > active₁ > U101₃ > ok₁ > mark
proper₁ > U102₁ > active₁ > U103₃ > ok₁ > mark
proper₁ > U102₁ > active₁ > U104₂ > ok₁ > mark
proper₁ > U102₁ > active₁ > x₂ > ok₁ > mark
proper₁ > U102₁ > active₁ > U12₂ > ok₁ > mark
proper₁ > U102₁ > active₁ > U13₃ > ok₁ > mark
proper₁ > U102₁ > active₁ > U16₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U16₁ > tt > mark
proper₁ > U102₁ > active₁ > U23₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U23₁ > tt > mark
proper₁ > U102₁ > active₁ > U35₂ > ok₁ > mark
proper₁ > U102₁ > active₁ > U36₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U36₁ > tt > mark
proper₁ > U102₁ > active₁ > U41₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U42₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U42₁ > tt > mark
proper₁ > U102₁ > active₁ > U62₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U62₁ > tt > mark
proper₁ > U102₁ > active₁ > U71₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U72₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U81₃ > ok₁ > mark
proper₁ > U102₁ > active₁ > U82₃ > ok₁ > mark
proper₁ > U102₁ > active₁ > U83₃ > ok₁ > mark
proper₁ > U102₁ > active₁ > s₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U92₁ > ok₁ > mark
proper₁ > U102₁ > active₁ > U92₁ > 0 > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U101₃ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U103₃ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U104₂ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > x₂ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U12₂ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U13₃ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U16₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U16₁ > tt > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U23₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U23₁ > tt > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U35₂ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U36₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U36₁ > tt > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U41₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U42₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U42₁ > tt > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U62₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U62₁ > tt > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U71₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U72₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U81₃ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U82₃ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U83₃ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > s₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U92₁ > ok₁ > mark
proper₁ > isNat₁ > U31₂ > U32₂ > active₁ > U92₁ > 0 > mark
proper₁ > isNat₁ > U31₂ > U32₂ > isNatKind₁ > tt > mark
proper₁ > isNat₁ > U31₂ > U32₂ > isNatKind₁ > U41₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U101₃ > ok₁ > mark
proper₁ > U14₂ > active₁ > U103₃ > ok₁ > mark
proper₁ > U14₂ > active₁ > U104₂ > ok₁ > mark
proper₁ > U14₂ > active₁ > x₂ > ok₁ > mark
proper₁ > U14₂ > active₁ > U12₂ > ok₁ > mark
proper₁ > U14₂ > active₁ > U13₃ > ok₁ > mark
proper₁ > U14₂ > active₁ > U16₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U16₁ > tt > mark
proper₁ > U14₂ > active₁ > U23₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U23₁ > tt > mark
proper₁ > U14₂ > active₁ > U35₂ > ok₁ > mark
proper₁ > U14₂ > active₁ > U36₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U36₁ > tt > mark
proper₁ > U14₂ > active₁ > U41₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U42₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U42₁ > tt > mark
proper₁ > U14₂ > active₁ > U62₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U62₁ > tt > mark
proper₁ > U14₂ > active₁ > U71₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U72₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U81₃ > ok₁ > mark
proper₁ > U14₂ > active₁ > U82₃ > ok₁ > mark
proper₁ > U14₂ > active₁ > U83₃ > ok₁ > mark
proper₁ > U14₂ > active₁ > s₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U92₁ > ok₁ > mark
proper₁ > U14₂ > active₁ > U92₁ > 0 > mark
proper₁ > U34₁ > active₁ > U101₃ > ok₁ > mark
proper₁ > U34₁ > active₁ > U103₃ > ok₁ > mark
proper₁ > U34₁ > active₁ > U104₂ > ok₁ > mark
proper₁ > U34₁ > active₁ > x₂ > ok₁ > mark
proper₁ > U34₁ > active₁ > U12₂ > ok₁ > mark
proper₁ > U34₁ > active₁ > U13₃ > ok₁ > mark
proper₁ > U34₁ > active₁ > U16₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U16₁ > tt > mark
proper₁ > U34₁ > active₁ > U23₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U23₁ > tt > mark
proper₁ > U34₁ > active₁ > U35₂ > ok₁ > mark
proper₁ > U34₁ > active₁ > U36₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U36₁ > tt > mark
proper₁ > U34₁ > active₁ > U41₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U42₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U42₁ > tt > mark
proper₁ > U34₁ > active₁ > U62₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U62₁ > tt > mark
proper₁ > U34₁ > active₁ > U71₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U72₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U81₃ > ok₁ > mark
proper₁ > U34₁ > active₁ > U82₃ > ok₁ > mark
proper₁ > U34₁ > active₁ > U83₃ > ok₁ > mark
proper₁ > U34₁ > active₁ > s₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U92₁ > ok₁ > mark
proper₁ > U34₁ > active₁ > U92₁ > 0 > mark
proper₁ > U61₁ > active₁ > U101₃ > ok₁ > mark
proper₁ > U61₁ > active₁ > U103₃ > ok₁ > mark
proper₁ > U61₁ > active₁ > U104₂ > ok₁ > mark
proper₁ > U61₁ > active₁ > x₂ > ok₁ > mark
proper₁ > U61₁ > active₁ > U12₂ > ok₁ > mark
proper₁ > U61₁ > active₁ > U13₃ > ok₁ > mark
proper₁ > U61₁ > active₁ > U16₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U16₁ > tt > mark
proper₁ > U61₁ > active₁ > U23₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U23₁ > tt > mark
proper₁ > U61₁ > active₁ > U35₂ > ok₁ > mark
proper₁ > U61₁ > active₁ > U36₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U36₁ > tt > mark
proper₁ > U61₁ > active₁ > U41₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U42₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U42₁ > tt > mark
proper₁ > U61₁ > active₁ > U62₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U62₁ > tt > mark
proper₁ > U61₁ > active₁ > U71₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U72₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U81₃ > ok₁ > mark
proper₁ > U61₁ > active₁ > U82₃ > ok₁ > mark
proper₁ > U61₁ > active₁ > U83₃ > ok₁ > mark
proper₁ > U61₁ > active₁ > s₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U92₁ > ok₁ > mark
proper₁ > U61₁ > active₁ > U92₁ > 0 > mark
proper₁ > U61₁ > isNatKind₁ > tt > mark
proper₁ > U61₁ > isNatKind₁ > U41₁ > ok₁ > mark
top > active₁ > U101₃ > ok₁ > mark
top > active₁ > U103₃ > ok₁ > mark
top > active₁ > U104₂ > ok₁ > mark
top > active₁ > x₂ > ok₁ > mark
top > active₁ > U12₂ > ok₁ > mark
top > active₁ > U13₃ > ok₁ > mark
top > active₁ > U16₁ > ok₁ > mark
top > active₁ > U16₁ > tt > mark
top > active₁ > U23₁ > ok₁ > mark
top > active₁ > U23₁ > tt > mark
top > active₁ > U35₂ > ok₁ > mark
top > active₁ > U36₁ > ok₁ > mark
top > active₁ > U36₁ > tt > mark
top > active₁ > U41₁ > ok₁ > mark
top > active₁ > U42₁ > ok₁ > mark
top > active₁ > U42₁ > tt > mark
top > active₁ > U62₁ > ok₁ > mark
top > active₁ > U62₁ > tt > mark
top > active₁ > U71₁ > ok₁ > mark
top > active₁ > U72₁ > ok₁ > mark
top > active₁ > U81₃ > ok₁ > mark
top > active₁ > U82₃ > ok₁ > mark
top > active₁ > U83₃ > ok₁ > mark
top > active₁ > s₁ > ok₁ > mark
top > active₁ > U92₁ > ok₁ > mark
top > active₁ > U92₁ > 0 > mark

The following usable rules [FROCOS05] were oriented:

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

(54) Obligation:

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

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

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

(55) PisEmptyProof (EQUIVALENT transformation)

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

(56) TRUE

(57) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U81^1₃ > mark₁
active₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > U41₁ > U42₁ > mark₁
active₁ > isNatKind > U51₁ > tt > mark₁
active₁ > isNatKind > U61₁ > mark₁
active₁ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U11₃ > mark₁
active₁ > U104₃ > plus₂ > U41₁ > U42₁ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U12₃ > U13₃ > mark₁
active₁ > U14₃ > U15₂ > mark₁
active₁ > U16₁ > tt > mark₁
active₁ > U21₂ > U22₂ > mark₁
active₁ > U23₁ > tt > mark₁
active₁ > U31₃ > U32₃ > mark₁
active₁ > U33₃ > U34₃ > mark₁
active₁ > U35₂ > mark₁
active₁ > U36₁ > tt > mark₁
active₁ > U62₁ > tt > mark₁
active₁ > U82₃ > mark₁
active₁ > U83₃ > U84₃ > s₁ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U71₂ > U72₂ > mark₁
active₁ > 0 > U91₂ > U92₁ > mark₁
proper₁ > U101₃ > U102₃ > mark₁
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U11₃ > mark₁
proper₁ > U104₃ > plus₂ > U41₁ > U42₁ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U12₃ > U13₃ > mark₁
proper₁ > U14₃ > U15₂ > mark₁
proper₁ > U16₁ > tt > mark₁
proper₁ > U21₂ > U22₂ > mark₁
proper₁ > U23₁ > tt > mark₁
proper₁ > U31₃ > U32₃ > mark₁
proper₁ > U33₃ > U34₃ > mark₁
proper₁ > U35₂ > mark₁
proper₁ > U36₁ > tt > mark₁
proper₁ > U51₁ > tt > mark₁
proper₁ > U61₁ > mark₁
proper₁ > U62₁ > tt > mark₁
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U82₃ > mark₁
proper₁ > U83₃ > U84₃ > s₁ > mark₁
proper₁ > U91₂ > U92₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(59) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1, x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x2)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  x3
plus(x1, x2)  =  x2
x(x1, x2)  =  x2
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x2
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x2)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x2
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x2, x3)
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  x2
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x2)
U72(x1, x2)  =  x1
U81(x1, x2, x3)  =  x2
U82(x1, x2, x3)  =  x3
U83(x1, x2, x3)  =  x2
U84(x1, x2, x3)  =  x1
s(x1)  =  x1
U91(x1, x2)  =  U91(x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > U101₁ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > U101₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U103₂ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > U103₂ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U11₂ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > U11₂ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U15₁ > ok₁ > U81^1₂ > mark
proper₁ > U15₁ > ok₁ > active₁ > tt > mark
proper₁ > U31₂ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > U31₂ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U35₁ > ok₁ > U81^1₂ > mark
proper₁ > U35₁ > ok₁ > active₁ > tt > mark
proper₁ > U41₁ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > U41₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U71₁ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > U71₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > 0 > U91₁ > isNatKind₁ > ok₁ > U81^1₂ > mark
proper₁ > 0 > U91₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
top > active₁ > tt > mark

The following usable rules [FROCOS05] were oriented:

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

(61) Obligation:

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

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

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

(62) PisEmptyProof (EQUIVALENT transformation)

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

(63) TRUE

(64) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(ok(X1), ok(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x3)
tt  =  tt
U102(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x2)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x2, x3)
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  x1
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x2)
U62(x1)  =  x1
U71(x1, x2)  =  x1
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x3)
U84(x1, x2, x3)  =  U84(x1)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₁ > mark > x₂ > ok₁
active₁ > U101₁ > mark > U12₃ > ok₁
active₁ > U101₁ > mark > U15₁ > ok₁
active₁ > U101₁ > mark > U16₁ > ok₁
active₁ > U101₁ > mark > U21₂ > ok₁
active₁ > U101₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U101₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U101₁ > mark > U91₂ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U101₁ > isNatKind₁ > tt > 0
active₁ > U103₁ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U103₁ > isNatKind₁ > tt > 0
active₁ > U103₁ > U104₂ > mark > x₂ > ok₁
active₁ > U103₁ > U104₂ > mark > U12₃ > ok₁
active₁ > U103₁ > U104₂ > mark > U15₁ > ok₁
active₁ > U103₁ > U104₂ > mark > U16₁ > ok₁
active₁ > U103₁ > U104₂ > mark > U21₂ > ok₁
active₁ > U103₁ > U104₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U103₁ > U104₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U103₁ > U104₂ > mark > U91₂ > ok₁
active₁ > U13₁ > mark > x₂ > ok₁
active₁ > U13₁ > mark > U12₃ > ok₁
active₁ > U13₁ > mark > U15₁ > ok₁
active₁ > U13₁ > mark > U16₁ > ok₁
active₁ > U13₁ > mark > U21₂ > ok₁
active₁ > U13₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U13₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U13₁ > mark > U91₂ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U13₁ > isNatKind₁ > tt > 0
active₁ > U13₁ > U14₃ > isNat₁ > ok₁
active₁ > U13₁ > U14₃ > U15₁ > ok₁
active₁ > U22₁ > mark > x₂ > ok₁
active₁ > U22₁ > mark > U12₃ > ok₁
active₁ > U22₁ > mark > U15₁ > ok₁
active₁ > U22₁ > mark > U16₁ > ok₁
active₁ > U22₁ > mark > U21₂ > ok₁
active₁ > U22₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U22₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U22₁ > mark > U91₂ > ok₁
active₁ > U22₁ > isNat₁ > ok₁
active₁ > U23₁ > mark > x₂ > ok₁
active₁ > U23₁ > mark > U12₃ > ok₁
active₁ > U23₁ > mark > U15₁ > ok₁
active₁ > U23₁ > mark > U16₁ > ok₁
active₁ > U23₁ > mark > U21₂ > ok₁
active₁ > U23₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U23₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U23₁ > mark > U91₂ > ok₁
active₁ > U23₁ > tt > isNat₁ > ok₁
active₁ > U23₁ > tt > x₂ > ok₁
active₁ > U23₁ > tt > U12₃ > ok₁
active₁ > U23₁ > tt > U15₁ > ok₁
active₁ > U23₁ > tt > U16₁ > ok₁
active₁ > U23₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U23₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U23₁ > tt > 0
active₁ > U31₂ > mark > x₂ > ok₁
active₁ > U31₂ > mark > U12₃ > ok₁
active₁ > U31₂ > mark > U15₁ > ok₁
active₁ > U31₂ > mark > U16₁ > ok₁
active₁ > U31₂ > mark > U21₂ > ok₁
active₁ > U31₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U31₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U31₂ > mark > U91₂ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U31₂ > isNatKind₁ > tt > 0
active₁ > U35₁ > mark > x₂ > ok₁
active₁ > U35₁ > mark > U12₃ > ok₁
active₁ > U35₁ > mark > U15₁ > ok₁
active₁ > U35₁ > mark > U16₁ > ok₁
active₁ > U35₁ > mark > U21₂ > ok₁
active₁ > U35₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U35₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U35₁ > mark > U91₂ > ok₁
active₁ > U35₁ > isNat₁ > ok₁
active₁ > U41₁ > mark > x₂ > ok₁
active₁ > U41₁ > mark > U12₃ > ok₁
active₁ > U41₁ > mark > U15₁ > ok₁
active₁ > U41₁ > mark > U16₁ > ok₁
active₁ > U41₁ > mark > U21₂ > ok₁
active₁ > U41₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U41₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U41₁ > mark > U91₂ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U41₁ > isNatKind₁ > tt > 0
active₁ > U51₁ > tt > isNat₁ > ok₁
active₁ > U51₁ > tt > x₂ > ok₁
active₁ > U51₁ > tt > U12₃ > ok₁
active₁ > U51₁ > tt > U15₁ > ok₁
active₁ > U51₁ > tt > U16₁ > ok₁
active₁ > U51₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U51₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U51₁ > tt > 0
active₁ > U61₁ > mark > x₂ > ok₁
active₁ > U61₁ > mark > U12₃ > ok₁
active₁ > U61₁ > mark > U15₁ > ok₁
active₁ > U61₁ > mark > U16₁ > ok₁
active₁ > U61₁ > mark > U21₂ > ok₁
active₁ > U61₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U61₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U61₁ > mark > U91₂ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U61₁ > isNatKind₁ > tt > 0
active₁ > U72₂ > mark > x₂ > ok₁
active₁ > U72₂ > mark > U12₃ > ok₁
active₁ > U72₂ > mark > U15₁ > ok₁
active₁ > U72₂ > mark > U16₁ > ok₁
active₁ > U72₂ > mark > U21₂ > ok₁
active₁ > U72₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U72₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U72₂ > mark > U91₂ > ok₁
active₁ > U81₂ > mark > x₂ > ok₁
active₁ > U81₂ > mark > U12₃ > ok₁
active₁ > U81₂ > mark > U15₁ > ok₁
active₁ > U81₂ > mark > U16₁ > ok₁
active₁ > U81₂ > mark > U21₂ > ok₁
active₁ > U81₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U81₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U81₂ > mark > U91₂ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > isNat₁ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > x₂ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > U12₃ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > U15₁ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > U16₁ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
active₁ > U81₂ > isNatKind₁ > tt > 0
proper₁ > U101₁ > mark > x₂ > ok₁
proper₁ > U101₁ > mark > U12₃ > ok₁
proper₁ > U101₁ > mark > U15₁ > ok₁
proper₁ > U101₁ > mark > U16₁ > ok₁
proper₁ > U101₁ > mark > U21₂ > ok₁
proper₁ > U101₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U101₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U101₁ > mark > U91₂ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U101₁ > isNatKind₁ > tt > 0
proper₁ > U103₁ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U103₁ > isNatKind₁ > tt > 0
proper₁ > U103₁ > U104₂ > mark > x₂ > ok₁
proper₁ > U103₁ > U104₂ > mark > U12₃ > ok₁
proper₁ > U103₁ > U104₂ > mark > U15₁ > ok₁
proper₁ > U103₁ > U104₂ > mark > U16₁ > ok₁
proper₁ > U103₁ > U104₂ > mark > U21₂ > ok₁
proper₁ > U103₁ > U104₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U103₁ > U104₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U103₁ > U104₂ > mark > U91₂ > ok₁
proper₁ > U13₁ > mark > x₂ > ok₁
proper₁ > U13₁ > mark > U12₃ > ok₁
proper₁ > U13₁ > mark > U15₁ > ok₁
proper₁ > U13₁ > mark > U16₁ > ok₁
proper₁ > U13₁ > mark > U21₂ > ok₁
proper₁ > U13₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U13₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U13₁ > mark > U91₂ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U13₁ > isNatKind₁ > tt > 0
proper₁ > U13₁ > U14₃ > isNat₁ > ok₁
proper₁ > U13₁ > U14₃ > U15₁ > ok₁
proper₁ > U22₁ > mark > x₂ > ok₁
proper₁ > U22₁ > mark > U12₃ > ok₁
proper₁ > U22₁ > mark > U15₁ > ok₁
proper₁ > U22₁ > mark > U16₁ > ok₁
proper₁ > U22₁ > mark > U21₂ > ok₁
proper₁ > U22₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U22₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U22₁ > mark > U91₂ > ok₁
proper₁ > U22₁ > isNat₁ > ok₁
proper₁ > U23₁ > mark > x₂ > ok₁
proper₁ > U23₁ > mark > U12₃ > ok₁
proper₁ > U23₁ > mark > U15₁ > ok₁
proper₁ > U23₁ > mark > U16₁ > ok₁
proper₁ > U23₁ > mark > U21₂ > ok₁
proper₁ > U23₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U23₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U23₁ > mark > U91₂ > ok₁
proper₁ > U23₁ > tt > isNat₁ > ok₁
proper₁ > U23₁ > tt > x₂ > ok₁
proper₁ > U23₁ > tt > U12₃ > ok₁
proper₁ > U23₁ > tt > U15₁ > ok₁
proper₁ > U23₁ > tt > U16₁ > ok₁
proper₁ > U23₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U23₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U23₁ > tt > 0
proper₁ > U31₂ > mark > x₂ > ok₁
proper₁ > U31₂ > mark > U12₃ > ok₁
proper₁ > U31₂ > mark > U15₁ > ok₁
proper₁ > U31₂ > mark > U16₁ > ok₁
proper₁ > U31₂ > mark > U21₂ > ok₁
proper₁ > U31₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U31₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U31₂ > mark > U91₂ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U31₂ > isNatKind₁ > tt > 0
proper₁ > U35₁ > mark > x₂ > ok₁
proper₁ > U35₁ > mark > U12₃ > ok₁
proper₁ > U35₁ > mark > U15₁ > ok₁
proper₁ > U35₁ > mark > U16₁ > ok₁
proper₁ > U35₁ > mark > U21₂ > ok₁
proper₁ > U35₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U35₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U35₁ > mark > U91₂ > ok₁
proper₁ > U35₁ > isNat₁ > ok₁
proper₁ > U41₁ > mark > x₂ > ok₁
proper₁ > U41₁ > mark > U12₃ > ok₁
proper₁ > U41₁ > mark > U15₁ > ok₁
proper₁ > U41₁ > mark > U16₁ > ok₁
proper₁ > U41₁ > mark > U21₂ > ok₁
proper₁ > U41₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U41₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U41₁ > mark > U91₂ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U41₁ > isNatKind₁ > tt > 0
proper₁ > U51₁ > tt > isNat₁ > ok₁
proper₁ > U51₁ > tt > x₂ > ok₁
proper₁ > U51₁ > tt > U12₃ > ok₁
proper₁ > U51₁ > tt > U15₁ > ok₁
proper₁ > U51₁ > tt > U16₁ > ok₁
proper₁ > U51₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U51₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U51₁ > tt > 0
proper₁ > U61₁ > mark > x₂ > ok₁
proper₁ > U61₁ > mark > U12₃ > ok₁
proper₁ > U61₁ > mark > U15₁ > ok₁
proper₁ > U61₁ > mark > U16₁ > ok₁
proper₁ > U61₁ > mark > U21₂ > ok₁
proper₁ > U61₁ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U61₁ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U61₁ > mark > U91₂ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U61₁ > isNatKind₁ > tt > 0
proper₁ > U72₂ > mark > x₂ > ok₁
proper₁ > U72₂ > mark > U12₃ > ok₁
proper₁ > U72₂ > mark > U15₁ > ok₁
proper₁ > U72₂ > mark > U16₁ > ok₁
proper₁ > U72₂ > mark > U21₂ > ok₁
proper₁ > U72₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U72₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U72₂ > mark > U91₂ > ok₁
proper₁ > U81₂ > mark > x₂ > ok₁
proper₁ > U81₂ > mark > U12₃ > ok₁
proper₁ > U81₂ > mark > U15₁ > ok₁
proper₁ > U81₂ > mark > U16₁ > ok₁
proper₁ > U81₂ > mark > U21₂ > ok₁
proper₁ > U81₂ > mark > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U81₂ > mark > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U81₂ > mark > U91₂ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > isNat₁ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > x₂ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > U12₃ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > U15₁ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > U16₁ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > plus₂ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > U82₃ > U83₂ > U84₁ > s₁ > ok₁
proper₁ > U81₂ > isNatKind₁ > tt > 0

The following usable rules [FROCOS05] were oriented:

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

(66) Obligation:

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

U72¹(mark(X1), X2) → U72¹(X1, X2)

The TRS R consists of the following rules:

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

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > tt > U104₃ > plus₂ > U11₃ > mark₁
active₁ > tt > U104₃ > plus₂ > U71₂ > mark₁
active₁ > tt > U104₃ > plus₂ > U81₃ > mark₁
active₁ > tt > x₂ > U101₃ > U102₃ > U103₃ > mark₁
active₁ > tt > x₂ > U31₃ > mark₁
active₁ > tt > U12₃ > U13₃ > mark₁
active₁ > tt > U14₃ > mark₁
active₁ > tt > U15₂ > mark₁
active₁ > tt > U16₁ > mark₁
active₁ > tt > U22₂ > mark₁
active₁ > tt > U23₁ > mark₁
active₁ > tt > U32₃ > mark₁
active₁ > tt > U33₃ > U34₃ > mark₁
active₁ > tt > U35₂ > mark₁
active₁ > tt > U36₁ > mark₁
active₁ > tt > U72₂ > mark₁
active₁ > tt > U82₃ > mark₁
active₁ > tt > U83₃ > U84₃ > mark₁
active₁ > isNatKind > U51₁ > mark₁
active₁ > U21₂ > U22₂ > mark₁
active₁ > 0 > U71₂ > mark₁
proper₁ > U104₃ > plus₂ > U11₃ > mark₁
proper₁ > U104₃ > plus₂ > U71₂ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > x₂ > U101₃ > U102₃ > U103₃ > mark₁
proper₁ > x₂ > U31₃ > mark₁
proper₁ > U12₃ > U13₃ > mark₁
proper₁ > U14₃ > mark₁
proper₁ > U15₂ > mark₁
proper₁ > U16₁ > mark₁
proper₁ > U21₂ > U22₂ > mark₁
proper₁ > U23₁ > mark₁
proper₁ > U32₃ > mark₁
proper₁ > U33₃ > U34₃ > mark₁
proper₁ > U35₂ > mark₁
proper₁ > U36₁ > mark₁
proper₁ > U51₁ > mark₁
proper₁ > U72₂ > mark₁
proper₁ > U82₃ > mark₁
proper₁ > U83₃ > U84₃ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(68) Obligation:

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

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

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

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > mark₁
active₁ > isNatKind > tt
active₁ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U13₁ > mark₁
active₁ > U15₁ > mark₁
active₁ > U23₁ > mark₁
active₁ > U23₁ > tt
active₁ > U33₁ > mark₁
active₁ > U34₁ > mark₁
active₁ > U62₁ > mark₁
active₁ > U62₁ > tt
active₁ > U71₂ > U72₂ > mark₁
active₁ > U82₃ > isNat > tt
active₁ > U82₃ > isNat > U11₁ > mark₁
active₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > 0 > tt
proper₁ > U101₃ > U102₃ > mark₁
proper₁ > isNatKind > mark₁
proper₁ > isNatKind > tt
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U13₁ > mark₁
proper₁ > U15₁ > mark₁
proper₁ > U23₁ > mark₁
proper₁ > U23₁ > tt
proper₁ > U33₁ > mark₁
proper₁ > U34₁ > mark₁
proper₁ > U62₁ > mark₁
proper₁ > U62₁ > tt
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U82₃ > isNat > tt
proper₁ > U82₃ > isNat > U11₁ > mark₁
proper₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
proper₁ > 0 > tt

The following usable rules [FROCOS05] were oriented:

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

(73) Obligation:

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

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

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

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U71^1₁ > mark₁
active₁ > U101₃ > ok₁ > mark₁
active₁ > U102₃ > ok₁ > mark₁
active₁ > U103₃ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
active₁ > plus₂ > U11₃ > ok₁ > mark₁
active₁ > plus₂ > U81₃ > ok₁ > mark₁
active₁ > U12₃ > ok₁ > mark₁
active₁ > U13₃ > U14₃ > ok₁ > mark₁
active₁ > U15₂ > ok₁ > mark₁
active₁ > U21₂ > ok₁ > mark₁
active₁ > U22₂ > ok₁ > mark₁
active₁ > U31₃ > U32₃ > ok₁ > mark₁
active₁ > U33₃ > ok₁ > mark₁
active₁ > U34₃ > U35₂ > ok₁ > mark₁
active₁ > U36₁ > ok₁ > mark₁
active₁ > U36₁ > tt > mark₁
active₁ > U41₂ > ok₁ > mark₁
active₁ > U62₁ > ok₁ > mark₁
active₁ > U62₁ > tt > mark₁
active₁ > U71₂ > U72₂ > ok₁ > mark₁
active₁ > U82₃ > ok₁ > mark₁
active₁ > U83₃ > ok₁ > mark₁
active₁ > U84₃ > ok₁ > mark₁
active₁ > U92₁ > ok₁ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U91₂ > ok₁ > mark₁
proper₁ > U101₃ > ok₁ > mark₁
proper₁ > U102₃ > ok₁ > mark₁
proper₁ > U103₃ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
proper₁ > plus₂ > U11₃ > ok₁ > mark₁
proper₁ > plus₂ > U81₃ > ok₁ > mark₁
proper₁ > U12₃ > ok₁ > mark₁
proper₁ > U13₃ > U14₃ > ok₁ > mark₁
proper₁ > U15₂ > ok₁ > mark₁
proper₁ > U21₂ > ok₁ > mark₁
proper₁ > U22₂ > ok₁ > mark₁
proper₁ > U31₃ > U32₃ > ok₁ > mark₁
proper₁ > U33₃ > ok₁ > mark₁
proper₁ > U34₃ > U35₂ > ok₁ > mark₁
proper₁ > U36₁ > ok₁ > mark₁
proper₁ > U36₁ > tt > mark₁
proper₁ > U41₂ > ok₁ > mark₁
proper₁ > U62₁ > ok₁ > mark₁
proper₁ > U62₁ > tt > mark₁
proper₁ > U71₂ > U72₂ > ok₁ > mark₁
proper₁ > U82₃ > ok₁ > mark₁
proper₁ > U83₃ > ok₁ > mark₁
proper₁ > U84₃ > ok₁ > mark₁
proper₁ > U92₁ > ok₁ > mark₁
proper₁ > 0 > tt > mark₁
proper₁ > 0 > U91₂ > ok₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(75) Obligation:

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

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

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

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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

U62¹(ok(X)) → U62¹(X)
U62¹(mark(X)) → U62¹(X)

The TRS R consists of the following rules:

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

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X)) → U62¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(80) Obligation:

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

U62¹(ok(X)) → U62¹(X)

The TRS R consists of the following rules:

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

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(ok(X)) → U62¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1)  =  U62¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U62^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(82) Obligation:

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

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

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

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) TRUE

(85) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X1), X2) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > mark₁
active₁ > isNatKind > tt
active₁ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U13₁ > mark₁
active₁ > U15₁ > mark₁
active₁ > U23₁ > mark₁
active₁ > U23₁ > tt
active₁ > U33₁ > mark₁
active₁ > U34₁ > mark₁
active₁ > U62₁ > mark₁
active₁ > U62₁ > tt
active₁ > U71₂ > U72₂ > mark₁
active₁ > U82₃ > isNat > tt
active₁ > U82₃ > isNat > U11₁ > mark₁
active₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > 0 > tt
proper₁ > U101₃ > U102₃ > mark₁
proper₁ > isNatKind > mark₁
proper₁ > isNatKind > tt
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U13₁ > mark₁
proper₁ > U15₁ > mark₁
proper₁ > U23₁ > mark₁
proper₁ > U23₁ > tt
proper₁ > U33₁ > mark₁
proper₁ > U34₁ > mark₁
proper₁ > U62₁ > mark₁
proper₁ > U62₁ > tt
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U82₃ > isNat > tt
proper₁ > U82₃ > isNat > U11₁ > mark₁
proper₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
proper₁ > 0 > tt

The following usable rules [FROCOS05] were oriented:

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

(87) Obligation:

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

U61¹(ok(X1), ok(X2)) → U61¹(X1, X2)

The TRS R consists of the following rules:

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

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(ok(X1), ok(X2)) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U61^1₁ > mark₁
active₁ > U101₃ > ok₁ > mark₁
active₁ > U102₃ > ok₁ > mark₁
active₁ > U103₃ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
active₁ > plus₂ > U11₃ > ok₁ > mark₁
active₁ > plus₂ > U81₃ > ok₁ > mark₁
active₁ > U12₃ > ok₁ > mark₁
active₁ > U13₃ > U14₃ > ok₁ > mark₁
active₁ > U15₂ > ok₁ > mark₁
active₁ > U21₂ > ok₁ > mark₁
active₁ > U22₂ > ok₁ > mark₁
active₁ > U31₃ > U32₃ > ok₁ > mark₁
active₁ > U33₃ > ok₁ > mark₁
active₁ > U34₃ > U35₂ > ok₁ > mark₁
active₁ > U36₁ > ok₁ > mark₁
active₁ > U36₁ > tt > mark₁
active₁ > U41₂ > ok₁ > mark₁
active₁ > U62₁ > ok₁ > mark₁
active₁ > U62₁ > tt > mark₁
active₁ > U71₂ > U72₂ > ok₁ > mark₁
active₁ > U82₃ > ok₁ > mark₁
active₁ > U83₃ > ok₁ > mark₁
active₁ > U84₃ > ok₁ > mark₁
active₁ > U92₁ > ok₁ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U91₂ > ok₁ > mark₁
proper₁ > U101₃ > ok₁ > mark₁
proper₁ > U102₃ > ok₁ > mark₁
proper₁ > U103₃ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
proper₁ > plus₂ > U11₃ > ok₁ > mark₁
proper₁ > plus₂ > U81₃ > ok₁ > mark₁
proper₁ > U12₃ > ok₁ > mark₁
proper₁ > U13₃ > U14₃ > ok₁ > mark₁
proper₁ > U15₂ > ok₁ > mark₁
proper₁ > U21₂ > ok₁ > mark₁
proper₁ > U22₂ > ok₁ > mark₁
proper₁ > U31₃ > U32₃ > ok₁ > mark₁
proper₁ > U33₃ > ok₁ > mark₁
proper₁ > U34₃ > U35₂ > ok₁ > mark₁
proper₁ > U36₁ > ok₁ > mark₁
proper₁ > U36₁ > tt > mark₁
proper₁ > U41₂ > ok₁ > mark₁
proper₁ > U62₁ > ok₁ > mark₁
proper₁ > U62₁ > tt > mark₁
proper₁ > U71₂ > U72₂ > ok₁ > mark₁
proper₁ > U82₃ > ok₁ > mark₁
proper₁ > U83₃ > ok₁ > mark₁
proper₁ > U84₃ > ok₁ > mark₁
proper₁ > U92₁ > ok₁ > mark₁
proper₁ > 0 > tt > mark₁
proper₁ > 0 > U91₂ > ok₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(89) Obligation:

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

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

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

(90) PisEmptyProof (EQUIVALENT transformation)

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

(91) TRUE

(92) Obligation:

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

U51¹(ok(X)) → U51¹(X)
U51¹(mark(X)) → U51¹(X)

The TRS R consists of the following rules:

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

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X)) → U51¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(94) Obligation:

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

U51¹(ok(X)) → U51¹(X)

The TRS R consists of the following rules:

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

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(ok(X)) → U51¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1)  =  U51¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U51^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(96) Obligation:

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

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

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

(97) PisEmptyProof (EQUIVALENT transformation)

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

(98) TRUE

(99) Obligation:

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

U42¹(ok(X)) → U42¹(X)
U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

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

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(101) Obligation:

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

U42¹(ok(X)) → U42¹(X)

The TRS R consists of the following rules:

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

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

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(ok(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1)  =  U42¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U42^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(103) Obligation:

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

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

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

(104) PisEmptyProof (EQUIVALENT transformation)

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

(105) TRUE

(106) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x3
U12(x1, x2, x3)  =  U12(x3)
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  U14(x1, x3)
U15(x1, x2)  =  x2
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x2, x3)
U32(x1, x2, x3)  =  U32(x3)
U33(x1, x2, x3)  =  U33(x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  U35(x2)
U36(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  x2
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x2)
U72(x1, x2)  =  U72(x2)
U81(x1, x2, x3)  =  U81(x2, x3)
U82(x1, x2, x3)  =  U82(x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > U101₃ > U102₃ > ok₁
top > active₁ > U104₂ > plus₂ > ok₁
top > active₁ > U104₂ > x₂ > ok₁
top > active₁ > U12₁ > isNatKind₁ > ok₁
top > active₁ > U21₁ > ok₁
top > active₁ > U23₁ > tt > U102₃ > ok₁
top > active₁ > U23₁ > tt > U103₃ > isNatKind₁ > ok₁
top > active₁ > U23₁ > tt > plus₂ > ok₁
top > active₁ > U23₁ > tt > x₂ > ok₁
top > active₁ > U23₁ > tt > U14₂ > ok₁
top > active₁ > U23₁ > tt > U16₁ > ok₁
top > active₁ > U23₁ > tt > U22₂ > ok₁
top > active₁ > U23₁ > tt > U33₁ > isNatKind₁ > ok₁
top > active₁ > U23₁ > tt > U35₁ > ok₁
top > active₁ > U23₁ > tt > U42₁ > ok₁
top > active₁ > U23₁ > tt > U72₁ > ok₁
top > active₁ > U23₁ > tt > U83₃ > ok₁
top > active₁ > U23₁ > tt > 0 > ok₁
top > active₁ > U31₂ > ok₁
top > active₁ > U32₁ > U33₁ > isNatKind₁ > ok₁
top > active₁ > U62₁ > ok₁
top > active₁ > U71₁ > isNatKind₁ > ok₁
top > active₁ > U71₁ > U72₁ > ok₁
top > active₁ > U81₂ > ok₁
top > active₁ > U82₂ > U83₃ > ok₁
top > active₁ > U84₃ > plus₂ > ok₁
top > active₁ > U91₂ > isNatKind₁ > ok₁
top > proper₁ > U101₃ > U102₃ > ok₁
top > proper₁ > U104₂ > plus₂ > ok₁
top > proper₁ > U104₂ > x₂ > ok₁
top > proper₁ > U12₁ > isNatKind₁ > ok₁
top > proper₁ > U21₁ > ok₁
top > proper₁ > U23₁ > tt > U102₃ > ok₁
top > proper₁ > U23₁ > tt > U103₃ > isNatKind₁ > ok₁
top > proper₁ > U23₁ > tt > plus₂ > ok₁
top > proper₁ > U23₁ > tt > x₂ > ok₁
top > proper₁ > U23₁ > tt > U14₂ > ok₁
top > proper₁ > U23₁ > tt > U16₁ > ok₁
top > proper₁ > U23₁ > tt > U22₂ > ok₁
top > proper₁ > U23₁ > tt > U33₁ > isNatKind₁ > ok₁
top > proper₁ > U23₁ > tt > U35₁ > ok₁
top > proper₁ > U23₁ > tt > U42₁ > ok₁
top > proper₁ > U23₁ > tt > U72₁ > ok₁
top > proper₁ > U23₁ > tt > U83₃ > ok₁
top > proper₁ > U23₁ > tt > 0 > ok₁
top > proper₁ > U31₂ > ok₁
top > proper₁ > U32₁ > U33₁ > isNatKind₁ > ok₁
top > proper₁ > U62₁ > ok₁
top > proper₁ > U71₁ > isNatKind₁ > ok₁
top > proper₁ > U71₁ > U72₁ > ok₁
top > proper₁ > U81₂ > ok₁
top > proper₁ > U82₂ > U83₃ > ok₁
top > proper₁ > U84₃ > plus₂ > ok₁
top > proper₁ > U91₂ > isNatKind₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(108) Obligation:

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

U41¹(mark(X1), X2) → U41¹(X1, X2)

The TRS R consists of the following rules:

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

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

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U41^1₁ > ok
active₁ > U101₃ > mark₁ > ok
active₁ > U104₃ > plus₂ > U71₂ > mark₁ > ok
active₁ > U104₃ > x₂ > mark₁ > ok
active₁ > U31₃ > mark₁ > ok
active₁ > U32₃ > mark₁ > ok
active₁ > U33₃ > mark₁ > ok
active₁ > U34₃ > isNat₁ > U11₃ > mark₁ > ok
active₁ > U34₃ > isNat₁ > U21₂ > mark₁ > ok
active₁ > U34₃ > U35₂ > mark₁ > ok
active₁ > U41₂ > mark₁ > ok
active₁ > U51₁ > tt > U102₃ > U103₃ > mark₁ > ok
active₁ > U51₁ > tt > plus₂ > U71₂ > mark₁ > ok
active₁ > U51₁ > tt > x₂ > mark₁ > ok
active₁ > U51₁ > tt > U12₃ > U13₃ > mark₁ > ok
active₁ > U51₁ > tt > U14₃ > U15₂ > mark₁ > ok
active₁ > U51₁ > tt > U16₁ > mark₁ > ok
active₁ > U51₁ > tt > U22₂ > U23₁ > mark₁ > ok
active₁ > U51₁ > tt > U35₂ > mark₁ > ok
active₁ > U51₁ > tt > U36₁ > mark₁ > ok
active₁ > U51₁ > tt > U72₂ > mark₁ > ok
active₁ > U51₁ > tt > U82₃ > mark₁ > ok
active₁ > U51₁ > tt > U83₃ > U84₃ > mark₁ > ok
active₁ > U51₁ > tt > U92₁ > mark₁ > ok
active₁ > U51₁ > tt > 0 > U71₂ > mark₁ > ok
active₁ > U51₁ > tt > 0 > U91₂ > mark₁ > ok
active₁ > U61₂ > mark₁ > ok
active₁ > U81₃ > U82₃ > mark₁ > ok
top > ok

The following usable rules [FROCOS05] were oriented:

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

(110) Obligation:

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

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

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

(111) PisEmptyProof (EQUIVALENT transformation)

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

(112) TRUE

(113) Obligation:

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

U36¹(ok(X)) → U36¹(X)
U36¹(mark(X)) → U36¹(X)

The TRS R consists of the following rules:

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

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U36¹(mark(X)) → U36¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U36¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(115) Obligation:

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

U36¹(ok(X)) → U36¹(X)

The TRS R consists of the following rules:

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

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

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U36¹(ok(X)) → U36¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U36¹(x1)  =  U36¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U36^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(117) Obligation:

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

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

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

(118) PisEmptyProof (EQUIVALENT transformation)

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

(119) TRUE

(120) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U35¹(mark(X1), X2) → U35¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U35¹(x1, x2)  =  U35¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U35^1₂ > mark₁
active₁ > U101₃ > U102₃ > mark₁
active₁ > U103₃ > mark₁
active₁ > U104₃ > mark₁
active₁ > x₂ > isNat₁ > tt > U102₃ > mark₁
active₁ > x₂ > isNat₁ > tt > U16₁ > mark₁
active₁ > x₂ > isNat₁ > tt > U34₃ > mark₁
active₁ > x₂ > isNat₁ > tt > U72₂ > mark₁
active₁ > x₂ > isNat₁ > tt > U92₁ > mark₁
active₁ > x₂ > isNat₁ > tt > 0 > mark₁
active₁ > U11₃ > U12₃ > mark₁
active₁ > U13₃ > mark₁
active₁ > U14₃ > mark₁
active₁ > U15₂ > U16₁ > mark₁
active₁ > U21₂ > U22₂ > mark₁
active₁ > U31₃ > mark₁
active₁ > U32₃ > mark₁
active₁ > U33₃ > U34₃ > mark₁
active₁ > U35₂ > mark₁
active₁ > U41₂ > mark₁
active₁ > U51₁ > mark₁
active₁ > U61₂ > U62₁ > tt > U102₃ > mark₁
active₁ > U61₂ > U62₁ > tt > U16₁ > mark₁
active₁ > U61₂ > U62₁ > tt > U34₃ > mark₁
active₁ > U61₂ > U62₁ > tt > U72₂ > mark₁
active₁ > U61₂ > U62₁ > tt > U92₁ > mark₁
active₁ > U61₂ > U62₁ > tt > 0 > mark₁
active₁ > U71₂ > U72₂ > mark₁
active₁ > U82₃ > U83₃ > mark₁
active₁ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > U91₂ > U92₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(122) Obligation:

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

U35¹(ok(X1), ok(X2)) → U35¹(X1, X2)

The TRS R consists of the following rules:

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

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U35¹(ok(X1), ok(X2)) → U35¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U35¹(x1, x2)  =  x2
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  x1
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  U12(x1, x3)
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  x3
U33(x1, x2, x3)  =  U33(x1, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  x2
U36(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  x2
U72(x1, x2)  =  U72(x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x2
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > U101₃ > U102₃ > U103₃ > ok₁
proper₁ > U101₃ > isNatKind₁ > ok₁
proper₁ > tt > U103₃ > ok₁
proper₁ > tt > U104₃ > plus₂ > ok₁
proper₁ > tt > U104₃ > x₂ > ok₁
proper₁ > tt > U12₂ > isNatKind₁ > ok₁
proper₁ > tt > U16₁ > ok₁
proper₁ > tt > U23₁ > ok₁
proper₁ > tt > U33₂ > ok₁
proper₁ > tt > U42₁ > ok₁
proper₁ > tt > U72₁ > ok₁
proper₁ > tt > U82₂ > ok₁
proper₁ > tt > U83₃ > ok₁
proper₁ > tt > U84₃ > plus₂ > ok₁
proper₁ > tt > U92₁ > ok₁
proper₁ > tt > 0 > ok₁
proper₁ > U11₂ > ok₁
proper₁ > U21₂ > ok₁
proper₁ > U31₃ > isNatKind₁ > ok₁
proper₁ > U61₂ > ok₁
proper₁ > U81₃ > ok₁
top > active₁ > U101₃ > U102₃ > U103₃ > ok₁
top > active₁ > U101₃ > isNatKind₁ > ok₁
top > active₁ > tt > U103₃ > ok₁
top > active₁ > tt > U104₃ > plus₂ > ok₁
top > active₁ > tt > U104₃ > x₂ > ok₁
top > active₁ > tt > U12₂ > isNatKind₁ > ok₁
top > active₁ > tt > U16₁ > ok₁
top > active₁ > tt > U23₁ > ok₁
top > active₁ > tt > U33₂ > ok₁
top > active₁ > tt > U42₁ > ok₁
top > active₁ > tt > U72₁ > ok₁
top > active₁ > tt > U82₂ > ok₁
top > active₁ > tt > U83₃ > ok₁
top > active₁ > tt > U84₃ > plus₂ > ok₁
top > active₁ > tt > U92₁ > ok₁
top > active₁ > tt > 0 > ok₁
top > active₁ > U11₂ > ok₁
top > active₁ > U21₂ > ok₁
top > active₁ > U31₃ > isNatKind₁ > ok₁
top > active₁ > U61₂ > ok₁
top > active₁ > U81₃ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(124) Obligation:

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

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

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

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  U34¹(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x2, x3)
U14(x1, x2, x3)  =  U14(x2, x3)
U15(x1, x2)  =  x2
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x3
U32(x1, x2, x3)  =  U32(x3)
U33(x1, x2, x3)  =  U33(x1, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  U35(x2)
U36(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x2
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x2, x3)
U82(x1, x2, x3)  =  U82(x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U102₃ > U103₃ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U103₃ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U14₂ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U23₁ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U32₁ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U33₂ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U42₁ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > U84₃ > ok₁ > U34^1₃
active₁ > isNat₁ > tt > 0
active₁ > isNat₁ > U11₂ > ok₁ > U34^1₃
active₁ > U104₂ > ok₁ > U34^1₃
active₁ > plus₂ > U11₂ > ok₁ > U34^1₃
active₁ > plus₂ > U81₂ > ok₁ > U34^1₃
active₁ > x₂ > U101₃ > ok₁ > U34^1₃
active₁ > x₂ > U91₂ > ok₁ > U34^1₃
active₁ > U12₃ > ok₁ > U34^1₃
active₁ > U13₂ > U14₂ > ok₁ > U34^1₃
active₁ > U35₁ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U103₃ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U14₂ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U23₁ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U32₁ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U33₂ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U42₁ > ok₁ > U34^1₃
active₁ > U51₁ > tt > U84₃ > ok₁ > U34^1₃
active₁ > U51₁ > tt > 0
active₁ > U62₁ > tt > U103₃ > ok₁ > U34^1₃
active₁ > U62₁ > tt > U14₂ > ok₁ > U34^1₃
active₁ > U62₁ > tt > U23₁ > ok₁ > U34^1₃
active₁ > U62₁ > tt > U32₁ > ok₁ > U34^1₃
active₁ > U62₁ > tt > U33₂ > ok₁ > U34^1₃
active₁ > U62₁ > tt > U42₁ > ok₁ > U34^1₃
active₁ > U62₁ > tt > U84₃ > ok₁ > U34^1₃
active₁ > U62₁ > tt > 0
active₁ > U71₂ > ok₁ > U34^1₃
active₁ > U72₂ > ok₁ > U34^1₃
active₁ > U82₂ > ok₁ > U34^1₃
active₁ > U83₂ > U84₃ > ok₁ > U34^1₃
active₁ > s₁ > U101₃ > ok₁ > U34^1₃
active₁ > s₁ > U81₂ > ok₁ > U34^1₃
active₁ > U92₁ > ok₁ > U34^1₃
active₁ > U92₁ > 0
proper₁ > U102₃ > U103₃ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U103₃ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U14₂ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U23₁ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U32₁ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U33₂ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U42₁ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > U84₃ > ok₁ > U34^1₃
proper₁ > isNat₁ > tt > 0
proper₁ > isNat₁ > U11₂ > ok₁ > U34^1₃
proper₁ > U104₂ > ok₁ > U34^1₃
proper₁ > plus₂ > U11₂ > ok₁ > U34^1₃
proper₁ > plus₂ > U81₂ > ok₁ > U34^1₃
proper₁ > x₂ > U101₃ > ok₁ > U34^1₃
proper₁ > x₂ > U91₂ > ok₁ > U34^1₃
proper₁ > U12₃ > ok₁ > U34^1₃
proper₁ > U13₂ > U14₂ > ok₁ > U34^1₃
proper₁ > U35₁ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U103₃ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U14₂ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U23₁ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U32₁ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U33₂ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U42₁ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > U84₃ > ok₁ > U34^1₃
proper₁ > U51₁ > tt > 0
proper₁ > U62₁ > tt > U103₃ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > U14₂ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > U23₁ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > U32₁ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > U33₂ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > U42₁ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > U84₃ > ok₁ > U34^1₃
proper₁ > U62₁ > tt > 0
proper₁ > U71₂ > ok₁ > U34^1₃
proper₁ > U72₂ > ok₁ > U34^1₃
proper₁ > U82₂ > ok₁ > U34^1₃
proper₁ > U83₂ > U84₃ > ok₁ > U34^1₃
proper₁ > s₁ > U101₃ > ok₁ > U34^1₃
proper₁ > s₁ > U81₂ > ok₁ > U34^1₃
proper₁ > U92₁ > ok₁ > U34^1₃
proper₁ > U92₁ > 0

The following usable rules [FROCOS05] were oriented:

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

(129) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(130) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U34¹(x1, x2, x3)  =  U34¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U34^1₃ > mark₁
active₁ > isNatKind > proper₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > proper₁ > U103₃ > U104₃ > x₂ > mark₁
active₁ > isNatKind > proper₁ > plus₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U11₃ > mark₁
active₁ > isNatKind > proper₁ > U12₃ > U13₃ > U14₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U21₂ > mark₁
active₁ > isNatKind > proper₁ > U31₃ > U32₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U102₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U104₃ > x₂ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U13₃ > U14₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U15₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U22₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U32₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U33₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U34₃ > U35₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U72₂ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U82₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U82₃ > U83₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U84₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > 0 > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U51₁ > mark₁
active₁ > isNatKind > proper₁ > U71₂ > U72₂ > mark₁
active₁ > isNatKind > proper₁ > U81₃ > U82₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U81₃ > U82₃ > U83₃ > mark₁
active₁ > isNatKind > proper₁ > U92₁ > 0 > isNat₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(131) Obligation:

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

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

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

(132) PisEmptyProof (EQUIVALENT transformation)

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

(133) TRUE

(134) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(135) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  U33¹(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x2, x3)
U14(x1, x2, x3)  =  U14(x2, x3)
U15(x1, x2)  =  x2
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x3
U32(x1, x2, x3)  =  U32(x3)
U33(x1, x2, x3)  =  U33(x1, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  U35(x2)
U36(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x2
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x2, x3)
U82(x1, x2, x3)  =  U82(x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U102₃ > U103₃ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U103₃ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U14₂ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U23₁ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U32₁ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U33₂ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U42₁ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > U84₃ > ok₁ > U33^1₃
active₁ > isNat₁ > tt > 0
active₁ > isNat₁ > U11₂ > ok₁ > U33^1₃
active₁ > U104₂ > ok₁ > U33^1₃
active₁ > plus₂ > U11₂ > ok₁ > U33^1₃
active₁ > plus₂ > U81₂ > ok₁ > U33^1₃
active₁ > x₂ > U101₃ > ok₁ > U33^1₃
active₁ > x₂ > U91₂ > ok₁ > U33^1₃
active₁ > U12₃ > ok₁ > U33^1₃
active₁ > U13₂ > U14₂ > ok₁ > U33^1₃
active₁ > U35₁ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U103₃ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U14₂ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U23₁ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U32₁ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U33₂ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U42₁ > ok₁ > U33^1₃
active₁ > U51₁ > tt > U84₃ > ok₁ > U33^1₃
active₁ > U51₁ > tt > 0
active₁ > U62₁ > tt > U103₃ > ok₁ > U33^1₃
active₁ > U62₁ > tt > U14₂ > ok₁ > U33^1₃
active₁ > U62₁ > tt > U23₁ > ok₁ > U33^1₃
active₁ > U62₁ > tt > U32₁ > ok₁ > U33^1₃
active₁ > U62₁ > tt > U33₂ > ok₁ > U33^1₃
active₁ > U62₁ > tt > U42₁ > ok₁ > U33^1₃
active₁ > U62₁ > tt > U84₃ > ok₁ > U33^1₃
active₁ > U62₁ > tt > 0
active₁ > U71₂ > ok₁ > U33^1₃
active₁ > U72₂ > ok₁ > U33^1₃
active₁ > U82₂ > ok₁ > U33^1₃
active₁ > U83₂ > U84₃ > ok₁ > U33^1₃
active₁ > s₁ > U101₃ > ok₁ > U33^1₃
active₁ > s₁ > U81₂ > ok₁ > U33^1₃
active₁ > U92₁ > ok₁ > U33^1₃
active₁ > U92₁ > 0
proper₁ > U102₃ > U103₃ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U103₃ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U14₂ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U23₁ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U32₁ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U33₂ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U42₁ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > U84₃ > ok₁ > U33^1₃
proper₁ > isNat₁ > tt > 0
proper₁ > isNat₁ > U11₂ > ok₁ > U33^1₃
proper₁ > U104₂ > ok₁ > U33^1₃
proper₁ > plus₂ > U11₂ > ok₁ > U33^1₃
proper₁ > plus₂ > U81₂ > ok₁ > U33^1₃
proper₁ > x₂ > U101₃ > ok₁ > U33^1₃
proper₁ > x₂ > U91₂ > ok₁ > U33^1₃
proper₁ > U12₃ > ok₁ > U33^1₃
proper₁ > U13₂ > U14₂ > ok₁ > U33^1₃
proper₁ > U35₁ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U103₃ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U14₂ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U23₁ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U32₁ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U33₂ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U42₁ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > U84₃ > ok₁ > U33^1₃
proper₁ > U51₁ > tt > 0
proper₁ > U62₁ > tt > U103₃ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > U14₂ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > U23₁ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > U32₁ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > U33₂ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > U42₁ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > U84₃ > ok₁ > U33^1₃
proper₁ > U62₁ > tt > 0
proper₁ > U71₂ > ok₁ > U33^1₃
proper₁ > U72₂ > ok₁ > U33^1₃
proper₁ > U82₂ > ok₁ > U33^1₃
proper₁ > U83₂ > U84₃ > ok₁ > U33^1₃
proper₁ > s₁ > U101₃ > ok₁ > U33^1₃
proper₁ > s₁ > U81₂ > ok₁ > U33^1₃
proper₁ > U92₁ > ok₁ > U33^1₃
proper₁ > U92₁ > 0

The following usable rules [FROCOS05] were oriented:

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

(136) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1, x2, x3)  =  U33¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U33^1₃ > mark₁
active₁ > isNatKind > proper₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > proper₁ > U103₃ > U104₃ > x₂ > mark₁
active₁ > isNatKind > proper₁ > plus₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U11₃ > mark₁
active₁ > isNatKind > proper₁ > U12₃ > U13₃ > U14₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U21₂ > mark₁
active₁ > isNatKind > proper₁ > U31₃ > U32₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U102₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U104₃ > x₂ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U13₃ > U14₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U15₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U22₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U32₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U33₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U34₃ > U35₂ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U72₂ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U82₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U82₃ > U83₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > U84₃ > mark₁
active₁ > isNatKind > proper₁ > U41₁ > U42₁ > tt > 0 > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U51₁ > mark₁
active₁ > isNatKind > proper₁ > U71₂ > U72₂ > mark₁
active₁ > isNatKind > proper₁ > U81₃ > U82₃ > isNat₁ > mark₁
active₁ > isNatKind > proper₁ > U81₃ > U82₃ > U83₃ > mark₁
active₁ > isNatKind > proper₁ > U92₁ > 0 > isNat₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(138) Obligation:

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

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

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

(139) PisEmptyProof (EQUIVALENT transformation)

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

(140) TRUE

(141) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  U32¹(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x3
U12(x1, x2, x3)  =  U12(x3)
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x3
U32(x1, x2, x3)  =  x3
U33(x1, x2, x3)  =  U33(x1, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  U35(x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x2
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x2)
U72(x1, x2)  =  x2
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > ok₁
active₁ > U101₃ > isNatKind₁ > U41₁ > ok₁
active₁ > U101₃ > isNatKind₁ > U51₁ > ok₁
active₁ > tt > U102₃ > ok₁
active₁ > tt > isNatKind₁ > U41₁ > ok₁
active₁ > tt > isNatKind₁ > U51₁ > ok₁
active₁ > tt > U104₃ > x₂ > ok₁
active₁ > tt > U23₁ > ok₁
active₁ > tt > U33₂ > ok₁
active₁ > tt > U35₁ > ok₁
active₁ > tt > U36₁ > ok₁
active₁ > tt > U42₁ > ok₁
active₁ > tt > U62₁ > ok₁
active₁ > tt > U82₃ > U83₂ > ok₁
active₁ > tt > s₁ > U51₁ > ok₁
active₁ > U103₃ > isNatKind₁ > U41₁ > ok₁
active₁ > U103₃ > isNatKind₁ > U51₁ > ok₁
active₁ > U103₃ > U104₃ > x₂ > ok₁
active₁ > plus₂ > isNatKind₁ > U41₁ > ok₁
active₁ > plus₂ > isNatKind₁ > U51₁ > ok₁
active₁ > plus₂ > U81₃ > U82₃ > U83₂ > ok₁
active₁ > U12₁ > isNatKind₁ > U41₁ > ok₁
active₁ > U12₁ > isNatKind₁ > U51₁ > ok₁
active₁ > U71₁ > isNatKind₁ > U41₁ > ok₁
active₁ > U71₁ > isNatKind₁ > U51₁ > ok₁
active₁ > U84₂ > s₁ > U51₁ > ok₁
active₁ > 0 > U91₂ > ok₁
proper₁ > U101₃ > U102₃ > ok₁
proper₁ > U101₃ > isNatKind₁ > U41₁ > ok₁
proper₁ > U101₃ > isNatKind₁ > U51₁ > ok₁
proper₁ > tt > U102₃ > ok₁
proper₁ > tt > isNatKind₁ > U41₁ > ok₁
proper₁ > tt > isNatKind₁ > U51₁ > ok₁
proper₁ > tt > U104₃ > x₂ > ok₁
proper₁ > tt > U23₁ > ok₁
proper₁ > tt > U33₂ > ok₁
proper₁ > tt > U35₁ > ok₁
proper₁ > tt > U36₁ > ok₁
proper₁ > tt > U42₁ > ok₁
proper₁ > tt > U62₁ > ok₁
proper₁ > tt > U82₃ > U83₂ > ok₁
proper₁ > tt > s₁ > U51₁ > ok₁
proper₁ > U103₃ > isNatKind₁ > U41₁ > ok₁
proper₁ > U103₃ > isNatKind₁ > U51₁ > ok₁
proper₁ > U103₃ > U104₃ > x₂ > ok₁
proper₁ > plus₂ > isNatKind₁ > U41₁ > ok₁
proper₁ > plus₂ > isNatKind₁ > U51₁ > ok₁
proper₁ > plus₂ > U81₃ > U82₃ > U83₂ > ok₁
proper₁ > U12₁ > isNatKind₁ > U41₁ > ok₁
proper₁ > U12₁ > isNatKind₁ > U51₁ > ok₁
proper₁ > U71₁ > isNatKind₁ > U41₁ > ok₁
proper₁ > U71₁ > isNatKind₁ > U51₁ > ok₁
proper₁ > U84₂ > s₁ > U51₁ > ok₁
proper₁ > 0 > U91₂ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(143) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U102₃ > U103₃ > mark₁
active₁ > U102₃ > isNat
active₁ > isNatKind > U41₁ > mark₁
active₁ > isNatKind > U51₁ > mark₁
active₁ > isNatKind > U51₁ > tt
active₁ > U104₃ > x₂ > U101₃ > mark₁
active₁ > U104₃ > x₂ > isNat
active₁ > plus₂ > U41₁ > mark₁
active₁ > plus₂ > U71₂ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > isNat
active₁ > U13₁ > mark₁
active₁ > U14₁ > mark₁
active₁ > U14₁ > isNat
active₁ > U22₁ > mark₁
active₁ > U22₁ > isNat
active₁ > U23₁ > mark₁
active₁ > U23₁ > tt
active₁ > U33₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > U83₃ > mark₁
active₁ > U84₃ > mark₁
active₁ > s₁ > U101₃ > mark₁
active₁ > s₁ > U51₁ > mark₁
active₁ > s₁ > U51₁ > tt
active₁ > s₁ > U81₃ > U82₃ > mark₁
active₁ > s₁ > U81₃ > U82₃ > isNat
active₁ > U92₁ > mark₁
active₁ > 0 > tt
active₁ > 0 > isNat
active₁ > 0 > U71₂ > mark₁
proper₁ > U102₃ > U103₃ > mark₁
proper₁ > U102₃ > isNat
proper₁ > isNatKind > U41₁ > mark₁
proper₁ > isNatKind > U51₁ > mark₁
proper₁ > isNatKind > U51₁ > tt
proper₁ > U104₃ > x₂ > U101₃ > mark₁
proper₁ > U104₃ > x₂ > isNat
proper₁ > plus₂ > U41₁ > mark₁
proper₁ > plus₂ > U71₂ > mark₁
proper₁ > plus₂ > U81₃ > U82₃ > mark₁
proper₁ > plus₂ > U81₃ > U82₃ > isNat
proper₁ > U13₁ > mark₁
proper₁ > U14₁ > mark₁
proper₁ > U14₁ > isNat
proper₁ > U22₁ > mark₁
proper₁ > U22₁ > isNat
proper₁ > U23₁ > mark₁
proper₁ > U23₁ > tt
proper₁ > U33₁ > mark₁
proper₁ > U72₂ > mark₁
proper₁ > U83₃ > mark₁
proper₁ > U84₃ > mark₁
proper₁ > s₁ > U101₃ > mark₁
proper₁ > s₁ > U51₁ > mark₁
proper₁ > s₁ > U51₁ > tt
proper₁ > s₁ > U81₃ > U82₃ > mark₁
proper₁ > s₁ > U81₃ > U82₃ > isNat
proper₁ > U92₁ > mark₁
proper₁ > 0 > tt
proper₁ > 0 > isNat
proper₁ > 0 > U71₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(145) Obligation:

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

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

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

(146) PisEmptyProof (EQUIVALENT transformation)

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

(147) TRUE

(148) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  U31¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > mark₁
active₁ > U11₃ > U12₃ > mark₁
active₁ > U31₃ > mark₁
active₁ > U41₂ > U42₁ > mark₁
active₁ > U81₃ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U103₃ > U104₃ > plus₂ > mark₁
active₁ > 0 > tt > isNat₁ > U21₂ > mark₁
active₁ > 0 > tt > x₂ > U61₂ > mark₁
active₁ > 0 > tt > U12₃ > mark₁
active₁ > 0 > tt > U13₃ > U14₃ > mark₁
active₁ > 0 > tt > U15₂ > mark₁
active₁ > 0 > tt > U22₂ > U23₁ > mark₁
active₁ > 0 > tt > U32₃ > mark₁
active₁ > 0 > tt > U33₃ > mark₁
active₁ > 0 > tt > U34₃ > mark₁
active₁ > 0 > tt > U35₂ > mark₁
active₁ > 0 > tt > U42₁ > mark₁
active₁ > 0 > tt > U72₂ > mark₁
active₁ > 0 > tt > U82₃ > U83₃ > mark₁
active₁ > 0 > tt > U84₃ > s₁ > mark₁
active₁ > 0 > U71₂ > mark₁
active₁ > 0 > U91₂ > mark₁
proper₁ > U101₃ > mark₁
proper₁ > U102₃ > mark₁
proper₁ > U103₃ > U104₃ > plus₂ > mark₁
proper₁ > isNat₁ > U21₂ > mark₁
proper₁ > x₂ > U61₂ > mark₁
proper₁ > U11₃ > U12₃ > mark₁
proper₁ > U13₃ > U14₃ > mark₁
proper₁ > U15₂ > mark₁
proper₁ > U22₂ > U23₁ > mark₁
proper₁ > U31₃ > mark₁
proper₁ > U32₃ > mark₁
proper₁ > U33₃ > mark₁
proper₁ > U34₃ > mark₁
proper₁ > U35₂ > mark₁
proper₁ > U41₂ > U42₁ > mark₁
proper₁ > U71₂ > mark₁
proper₁ > U72₂ > mark₁
proper₁ > U81₃ > mark₁
proper₁ > U82₃ > U83₃ > mark₁
proper₁ > U84₃ > s₁ > mark₁
proper₁ > U91₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(150) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(151) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2, x3)  =  U31¹(x2, x3)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1)
U72(x1, x2)  =  x1
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  x1
U83(x1, x2, x3)  =  U83(x1)
U84(x1, x2, x3)  =  U84(x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

0 > isNat₁ > tt > plus₂ > ok₁
0 > isNat₁ > tt > plus₂ > mark
0 > isNat₁ > tt > x₁ > U31₃ > ok₁
0 > isNat₁ > tt > x₁ > U31₃ > mark
0 > isNat₁ > tt > U16₁ > ok₁
0 > isNat₁ > tt > U16₁ > mark
0 > isNat₁ > tt > U33₃ > ok₁
0 > isNat₁ > tt > U33₃ > mark
0 > isNat₁ > tt > U42₁ > ok₁
0 > isNat₁ > tt > U42₁ > mark
0 > isNat₁ > tt > s₁ > ok₁
0 > isNat₁ > tt > s₁ > mark
0 > isNat₁ > U11₂ > ok₁
0 > isNat₁ > U11₂ > mark
0 > U71₁ > ok₁
0 > U71₁ > mark
0 > U91₂ > ok₁
0 > U91₂ > mark
proper₁ > U101₃ > ok₁
proper₁ > U101₃ > mark
proper₁ > U102₃ > ok₁
proper₁ > U102₃ > mark
proper₁ > U103₃ > U104₃ > plus₂ > ok₁
proper₁ > U103₃ > U104₃ > plus₂ > mark
proper₁ > U103₃ > U104₃ > x₁ > U31₃ > ok₁
proper₁ > U103₃ > U104₃ > x₁ > U31₃ > mark
proper₁ > U13₃ > ok₁
proper₁ > U13₃ > mark
proper₁ > U15₂ > isNat₁ > tt > plus₂ > ok₁
proper₁ > U15₂ > isNat₁ > tt > plus₂ > mark
proper₁ > U15₂ > isNat₁ > tt > x₁ > U31₃ > ok₁
proper₁ > U15₂ > isNat₁ > tt > x₁ > U31₃ > mark
proper₁ > U15₂ > isNat₁ > tt > U16₁ > ok₁
proper₁ > U15₂ > isNat₁ > tt > U16₁ > mark
proper₁ > U15₂ > isNat₁ > tt > U33₃ > ok₁
proper₁ > U15₂ > isNat₁ > tt > U33₃ > mark
proper₁ > U15₂ > isNat₁ > tt > U42₁ > ok₁
proper₁ > U15₂ > isNat₁ > tt > U42₁ > mark
proper₁ > U15₂ > isNat₁ > tt > s₁ > ok₁
proper₁ > U15₂ > isNat₁ > tt > s₁ > mark
proper₁ > U15₂ > isNat₁ > U11₂ > ok₁
proper₁ > U15₂ > isNat₁ > U11₂ > mark
proper₁ > U21₁ > ok₁
proper₁ > U21₁ > mark
proper₁ > U22₁ > ok₁
proper₁ > U22₁ > mark
proper₁ > U32₃ > U33₃ > ok₁
proper₁ > U32₃ > U33₃ > mark
proper₁ > U35₁ > ok₁
proper₁ > U35₁ > mark
proper₁ > U41₁ > U42₁ > ok₁
proper₁ > U41₁ > U42₁ > mark
proper₁ > U51₁ > tt > plus₂ > ok₁
proper₁ > U51₁ > tt > plus₂ > mark
proper₁ > U51₁ > tt > x₁ > U31₃ > ok₁
proper₁ > U51₁ > tt > x₁ > U31₃ > mark
proper₁ > U51₁ > tt > U16₁ > ok₁
proper₁ > U51₁ > tt > U16₁ > mark
proper₁ > U51₁ > tt > U33₃ > ok₁
proper₁ > U51₁ > tt > U33₃ > mark
proper₁ > U51₁ > tt > U42₁ > ok₁
proper₁ > U51₁ > tt > U42₁ > mark
proper₁ > U51₁ > tt > s₁ > ok₁
proper₁ > U51₁ > tt > s₁ > mark
proper₁ > U61₂ > ok₁
proper₁ > U61₂ > mark
proper₁ > U71₁ > ok₁
proper₁ > U71₁ > mark
proper₁ > U81₃ > ok₁
proper₁ > U81₃ > mark
proper₁ > U83₁ > U84₂ > plus₂ > ok₁
proper₁ > U83₁ > U84₂ > plus₂ > mark
proper₁ > U83₁ > U84₂ > s₁ > ok₁
proper₁ > U83₁ > U84₂ > s₁ > mark
proper₁ > U91₂ > ok₁
proper₁ > U91₂ > mark

The following usable rules [FROCOS05] were oriented:

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

(152) Obligation:

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

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

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

(153) PisEmptyProof (EQUIVALENT transformation)

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

(154) TRUE

(155) Obligation:

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

U23¹(ok(X)) → U23¹(X)
U23¹(mark(X)) → U23¹(X)

The TRS R consists of the following rules:

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

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

(156) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(mark(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(157) Obligation:

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

U23¹(ok(X)) → U23¹(X)

The TRS R consists of the following rules:

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

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

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(ok(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  U23¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U23^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(159) Obligation:

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

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

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

(160) PisEmptyProof (EQUIVALENT transformation)

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

(161) TRUE

(162) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(163) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > mark₁
active₁ > isNatKind > tt
active₁ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U13₁ > mark₁
active₁ > U15₁ > mark₁
active₁ > U23₁ > mark₁
active₁ > U23₁ > tt
active₁ > U33₁ > mark₁
active₁ > U34₁ > mark₁
active₁ > U62₁ > mark₁
active₁ > U62₁ > tt
active₁ > U71₂ > U72₂ > mark₁
active₁ > U82₃ > isNat > tt
active₁ > U82₃ > isNat > U11₁ > mark₁
active₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > 0 > tt
proper₁ > U101₃ > U102₃ > mark₁
proper₁ > isNatKind > mark₁
proper₁ > isNatKind > tt
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U13₁ > mark₁
proper₁ > U15₁ > mark₁
proper₁ > U23₁ > mark₁
proper₁ > U23₁ > tt
proper₁ > U33₁ > mark₁
proper₁ > U34₁ > mark₁
proper₁ > U62₁ > mark₁
proper₁ > U62₁ > tt
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U82₃ > isNat > tt
proper₁ > U82₃ > isNat > U11₁ > mark₁
proper₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
proper₁ > 0 > tt

The following usable rules [FROCOS05] were oriented:

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

(164) Obligation:

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

U22¹(ok(X1), ok(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

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

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

(165) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(ok(X1), ok(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U22^1₁ > mark₁
active₁ > U101₃ > ok₁ > mark₁
active₁ > U102₃ > ok₁ > mark₁
active₁ > U103₃ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
active₁ > plus₂ > U11₃ > ok₁ > mark₁
active₁ > plus₂ > U81₃ > ok₁ > mark₁
active₁ > U12₃ > ok₁ > mark₁
active₁ > U13₃ > U14₃ > ok₁ > mark₁
active₁ > U15₂ > ok₁ > mark₁
active₁ > U21₂ > ok₁ > mark₁
active₁ > U22₂ > ok₁ > mark₁
active₁ > U31₃ > U32₃ > ok₁ > mark₁
active₁ > U33₃ > ok₁ > mark₁
active₁ > U34₃ > U35₂ > ok₁ > mark₁
active₁ > U36₁ > ok₁ > mark₁
active₁ > U36₁ > tt > mark₁
active₁ > U41₂ > ok₁ > mark₁
active₁ > U62₁ > ok₁ > mark₁
active₁ > U62₁ > tt > mark₁
active₁ > U71₂ > U72₂ > ok₁ > mark₁
active₁ > U82₃ > ok₁ > mark₁
active₁ > U83₃ > ok₁ > mark₁
active₁ > U84₃ > ok₁ > mark₁
active₁ > U92₁ > ok₁ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U91₂ > ok₁ > mark₁
proper₁ > U101₃ > ok₁ > mark₁
proper₁ > U102₃ > ok₁ > mark₁
proper₁ > U103₃ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
proper₁ > plus₂ > U11₃ > ok₁ > mark₁
proper₁ > plus₂ > U81₃ > ok₁ > mark₁
proper₁ > U12₃ > ok₁ > mark₁
proper₁ > U13₃ > U14₃ > ok₁ > mark₁
proper₁ > U15₂ > ok₁ > mark₁
proper₁ > U21₂ > ok₁ > mark₁
proper₁ > U22₂ > ok₁ > mark₁
proper₁ > U31₃ > U32₃ > ok₁ > mark₁
proper₁ > U33₃ > ok₁ > mark₁
proper₁ > U34₃ > U35₂ > ok₁ > mark₁
proper₁ > U36₁ > ok₁ > mark₁
proper₁ > U36₁ > tt > mark₁
proper₁ > U41₂ > ok₁ > mark₁
proper₁ > U62₁ > ok₁ > mark₁
proper₁ > U62₁ > tt > mark₁
proper₁ > U71₂ > U72₂ > ok₁ > mark₁
proper₁ > U82₃ > ok₁ > mark₁
proper₁ > U83₃ > ok₁ > mark₁
proper₁ > U84₃ > ok₁ > mark₁
proper₁ > U92₁ > ok₁ > mark₁
proper₁ > 0 > tt > mark₁
proper₁ > 0 > U91₂ > ok₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(166) Obligation:

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

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

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

(167) PisEmptyProof (EQUIVALENT transformation)

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

(168) TRUE

(169) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(170) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > mark₁
active₁ > isNatKind > tt
active₁ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U13₁ > mark₁
active₁ > U15₁ > mark₁
active₁ > U23₁ > mark₁
active₁ > U23₁ > tt
active₁ > U33₁ > mark₁
active₁ > U34₁ > mark₁
active₁ > U62₁ > mark₁
active₁ > U62₁ > tt
active₁ > U71₂ > U72₂ > mark₁
active₁ > U82₃ > isNat > tt
active₁ > U82₃ > isNat > U11₁ > mark₁
active₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > 0 > tt
proper₁ > U101₃ > U102₃ > mark₁
proper₁ > isNatKind > mark₁
proper₁ > isNatKind > tt
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U13₁ > mark₁
proper₁ > U15₁ > mark₁
proper₁ > U23₁ > mark₁
proper₁ > U23₁ > tt
proper₁ > U33₁ > mark₁
proper₁ > U34₁ > mark₁
proper₁ > U62₁ > mark₁
proper₁ > U62₁ > tt
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U82₃ > isNat > tt
proper₁ > U82₃ > isNat > U11₁ > mark₁
proper₁ > U83₃ > U84₃ > plus₂ > U81₃ > mark₁
proper₁ > 0 > tt

The following usable rules [FROCOS05] were oriented:

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

(171) Obligation:

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

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

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

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

(172) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U21^1₁ > mark₁
active₁ > U101₃ > ok₁ > mark₁
active₁ > U102₃ > ok₁ > mark₁
active₁ > U103₃ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
active₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
active₁ > plus₂ > U11₃ > ok₁ > mark₁
active₁ > plus₂ > U81₃ > ok₁ > mark₁
active₁ > U12₃ > ok₁ > mark₁
active₁ > U13₃ > U14₃ > ok₁ > mark₁
active₁ > U15₂ > ok₁ > mark₁
active₁ > U21₂ > ok₁ > mark₁
active₁ > U22₂ > ok₁ > mark₁
active₁ > U31₃ > U32₃ > ok₁ > mark₁
active₁ > U33₃ > ok₁ > mark₁
active₁ > U34₃ > U35₂ > ok₁ > mark₁
active₁ > U36₁ > ok₁ > mark₁
active₁ > U36₁ > tt > mark₁
active₁ > U41₂ > ok₁ > mark₁
active₁ > U62₁ > ok₁ > mark₁
active₁ > U62₁ > tt > mark₁
active₁ > U71₂ > U72₂ > ok₁ > mark₁
active₁ > U82₃ > ok₁ > mark₁
active₁ > U83₃ > ok₁ > mark₁
active₁ > U84₃ > ok₁ > mark₁
active₁ > U92₁ > ok₁ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U91₂ > ok₁ > mark₁
proper₁ > U101₃ > ok₁ > mark₁
proper₁ > U102₃ > ok₁ > mark₁
proper₁ > U103₃ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U61₂ > ok₁ > mark₁
proper₁ > U104₃ > x₂ > U91₂ > ok₁ > mark₁
proper₁ > plus₂ > U11₃ > ok₁ > mark₁
proper₁ > plus₂ > U81₃ > ok₁ > mark₁
proper₁ > U12₃ > ok₁ > mark₁
proper₁ > U13₃ > U14₃ > ok₁ > mark₁
proper₁ > U15₂ > ok₁ > mark₁
proper₁ > U21₂ > ok₁ > mark₁
proper₁ > U22₂ > ok₁ > mark₁
proper₁ > U31₃ > U32₃ > ok₁ > mark₁
proper₁ > U33₃ > ok₁ > mark₁
proper₁ > U34₃ > U35₂ > ok₁ > mark₁
proper₁ > U36₁ > ok₁ > mark₁
proper₁ > U36₁ > tt > mark₁
proper₁ > U41₂ > ok₁ > mark₁
proper₁ > U62₁ > ok₁ > mark₁
proper₁ > U62₁ > tt > mark₁
proper₁ > U71₂ > U72₂ > ok₁ > mark₁
proper₁ > U82₃ > ok₁ > mark₁
proper₁ > U83₃ > ok₁ > mark₁
proper₁ > U84₃ > ok₁ > mark₁
proper₁ > U92₁ > ok₁ > mark₁
proper₁ > 0 > tt > mark₁
proper₁ > 0 > U91₂ > ok₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(173) Obligation:

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

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

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

(174) PisEmptyProof (EQUIVALENT transformation)

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

(175) TRUE

(176) Obligation:

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

U16¹(ok(X)) → U16¹(X)
U16¹(mark(X)) → U16¹(X)

The TRS R consists of the following rules:

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

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

(177) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U16¹(mark(X)) → U16¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U16¹(x1)  =  x1
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
active₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
active₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U14₁ > isNat > tt > U102₃ > mark₁
active₁ > U14₁ > isNat > tt > U22₁ > mark₁
active₁ > U15₁ > isNat > tt > U102₃ > mark₁
active₁ > U15₁ > isNat > tt > U22₁ > mark₁
active₁ > U23₁ > tt > U102₃ > mark₁
active₁ > U23₁ > tt > U22₁ > mark₁
active₁ > U32₁ > U33₁ > mark₁
active₁ > U35₁ > mark₁
active₁ > U36₁ > tt > U102₃ > mark₁
active₁ > U36₁ > tt > U22₁ > mark₁
active₁ > U51₁ > tt > U102₃ > mark₁
active₁ > U51₁ > tt > U22₁ > mark₁
active₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
active₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
active₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
active₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U22₁ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U103₃ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U103₃ > U104₃ > x₂ > U101₃ > mark₁
top > proper₁ > plus₂ > U81₃ > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U11₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U12₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U13₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U14₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U14₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U15₁ > isNat > tt > U102₃ > mark₁
top > proper₁ > U15₁ > isNat > tt > U22₁ > mark₁
top > proper₁ > U23₁ > tt > U102₃ > mark₁
top > proper₁ > U23₁ > tt > U22₁ > mark₁
top > proper₁ > U32₁ > U33₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U36₁ > tt > U102₃ > mark₁
top > proper₁ > U36₁ > tt > U22₁ > mark₁
top > proper₁ > U51₁ > tt > U102₃ > mark₁
top > proper₁ > U51₁ > tt > U22₁ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U61₁ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U102₃ > mark₁
top > proper₁ > U71₂ > isNatKind > tt > U22₁ > mark₁
top > proper₁ > U72₂ > mark₁
top > proper₁ > 0 > tt > U102₃ > mark₁
top > proper₁ > 0 > tt > U22₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(178) Obligation:

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

U16¹(ok(X)) → U16¹(X)

The TRS R consists of the following rules:

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

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

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U16¹(ok(X)) → U16¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U16¹(x1)  =  U16¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U16^1₁ > ok₁
top > active₁ > plus₂ > isNat₁ > ok₁
top > active₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > active₁ > plus₂ > U71₂ > mark₁ > ok₁
top > active₁ > plus₂ > U81₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > active₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > active₁ > x₂ > U31₃ > mark₁ > ok₁
top > active₁ > x₂ > U61₂ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > active₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > active₁ > U14₃ > mark₁ > ok₁
top > active₁ > U14₃ > isNat₁ > ok₁
top > active₁ > U15₂ > isNat₁ > ok₁
top > active₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > active₁ > U15₂ > U16₁ > mark₁ > ok₁
top > active₁ > U21₂ > mark₁ > ok₁
top > active₁ > U22₂ > mark₁ > ok₁
top > active₁ > U22₂ > isNat₁ > ok₁
top > active₁ > U32₃ > U33₃ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > mark₁ > ok₁
top > active₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > active₁ > U36₁ > tt > 0 > ok₁
top > active₁ > U36₁ > mark₁ > ok₁
top > active₁ > U41₂ > mark₁ > ok₁
top > active₁ > U42₁ > tt > 0 > ok₁
top > active₁ > U42₁ > mark₁ > ok₁
top > active₁ > U51₁ > tt > 0 > ok₁
top > active₁ > U51₁ > mark₁ > ok₁
top > active₁ > U62₁ > tt > 0 > ok₁
top > active₁ > U62₁ > mark₁ > ok₁
top > active₁ > U72₂ > mark₁ > ok₁
top > active₁ > U82₃ > mark₁ > ok₁
top > active₁ > U82₃ > isNat₁ > ok₁
top > active₁ > U83₃ > U84₃ > mark₁ > ok₁
top > proper₁ > plus₂ > isNat₁ > ok₁
top > proper₁ > plus₂ > U11₃ > U12₃ > U13₃ > mark₁ > ok₁
top > proper₁ > plus₂ > U71₂ > mark₁ > ok₁
top > proper₁ > plus₂ > U81₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
top > proper₁ > x₂ > U101₃ > U102₃ > isNat₁ > ok₁
top > proper₁ > x₂ > U31₃ > mark₁ > ok₁
top > proper₁ > x₂ > U61₂ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > mark₁ > ok₁
top > proper₁ > x₂ > U91₂ > U92₁ > 0 > ok₁
top > proper₁ > U14₃ > mark₁ > ok₁
top > proper₁ > U14₃ > isNat₁ > ok₁
top > proper₁ > U15₂ > isNat₁ > ok₁
top > proper₁ > U15₂ > U16₁ > tt > 0 > ok₁
top > proper₁ > U15₂ > U16₁ > mark₁ > ok₁
top > proper₁ > U21₂ > mark₁ > ok₁
top > proper₁ > U22₂ > mark₁ > ok₁
top > proper₁ > U22₂ > isNat₁ > ok₁
top > proper₁ > U32₃ > U33₃ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > mark₁ > ok₁
top > proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
top > proper₁ > U36₁ > tt > 0 > ok₁
top > proper₁ > U36₁ > mark₁ > ok₁
top > proper₁ > U41₂ > mark₁ > ok₁
top > proper₁ > U42₁ > tt > 0 > ok₁
top > proper₁ > U42₁ > mark₁ > ok₁
top > proper₁ > U51₁ > tt > 0 > ok₁
top > proper₁ > U51₁ > mark₁ > ok₁
top > proper₁ > U62₁ > tt > 0 > ok₁
top > proper₁ > U62₁ > mark₁ > ok₁
top > proper₁ > U72₂ > mark₁ > ok₁
top > proper₁ > U82₃ > mark₁ > ok₁
top > proper₁ > U82₃ > isNat₁ > ok₁
top > proper₁ > U83₃ > U84₃ > mark₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(180) Obligation:

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

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

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

(181) PisEmptyProof (EQUIVALENT transformation)

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

(182) TRUE

(183) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U15¹(mark(X1), X2) → U15¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U15¹(x1, x2)  =  U15¹(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  x1
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  U35(x1)
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > isNat > tt > isNatKind > proper₁ > U103₃ > U104₃ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > x₂ > U101₃ > U102₃ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U15₁ > U16₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U21₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U31₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U32₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U34₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U35₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U42₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U51₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U61₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U62₁ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U71₂ > U72₂ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U82₃ > U83₃ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U84₃ > s₁ > U81₃ > mark₁
active₁ > isNat > tt > isNatKind > proper₁ > U92₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U103₃ > U104₃ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > x₂ > U101₃ > U102₃ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U15₁ > U16₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U21₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U31₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U32₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U34₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U35₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U42₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U51₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U61₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U62₁ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U71₂ > U72₂ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U82₃ > U83₃ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U84₃ > s₁ > U81₃ > mark₁
active₁ > 0 > tt > isNatKind > proper₁ > U92₁ > mark₁
top > proper₁ > U103₃ > U104₃ > mark₁
top > proper₁ > x₂ > U101₃ > U102₃ > mark₁
top > proper₁ > U15₁ > U16₁ > mark₁
top > proper₁ > U21₁ > mark₁
top > proper₁ > U31₁ > mark₁
top > proper₁ > U32₁ > mark₁
top > proper₁ > U34₁ > mark₁
top > proper₁ > U35₁ > mark₁
top > proper₁ > U42₁ > mark₁
top > proper₁ > U51₁ > mark₁
top > proper₁ > U61₁ > mark₁
top > proper₁ > U62₁ > mark₁
top > proper₁ > U71₂ > U72₂ > mark₁
top > proper₁ > U82₃ > U83₃ > mark₁
top > proper₁ > U84₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U84₃ > s₁ > U81₃ > mark₁
top > proper₁ > U92₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(185) Obligation:

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

U15¹(ok(X1), ok(X2)) → U15¹(X1, X2)

The TRS R consists of the following rules:

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

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

(186) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U15¹(ok(X1), ok(X2)) → U15¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U15¹(x1, x2)  =  U15¹(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U15^1₁ > ok₁
active₁ > tt > U102₃ > mark₁ > ok₁
active₁ > tt > U103₃ > mark₁ > ok₁
active₁ > tt > U104₃ > plus₂ > mark₁ > ok₁
active₁ > tt > U104₃ > x₂ > U101₃ > mark₁ > ok₁
active₁ > tt > U104₃ > x₂ > U61₂ > mark₁ > ok₁
active₁ > tt > U12₃ > isNatKind₁ > U51₁ > mark₁ > ok₁
active₁ > tt > U12₃ > isNatKind₁ > U61₂ > mark₁ > ok₁
active₁ > tt > U12₃ > U13₃ > mark₁ > ok₁
active₁ > tt > U14₃ > U15₂ > mark₁ > ok₁
active₁ > tt > U16₁ > mark₁ > ok₁
active₁ > tt > U22₂ > mark₁ > ok₁
active₁ > tt > U23₁ > mark₁ > ok₁
active₁ > tt > U32₃ > mark₁ > ok₁
active₁ > tt > U33₃ > U34₃ > U35₂ > mark₁ > ok₁
active₁ > tt > U42₁ > mark₁ > ok₁
active₁ > tt > U62₁ > mark₁ > ok₁
active₁ > tt > U72₂ > mark₁ > ok₁
active₁ > tt > U82₃ > mark₁ > ok₁
active₁ > tt > U83₃ > mark₁ > ok₁
active₁ > tt > U84₃ > plus₂ > mark₁ > ok₁
active₁ > tt > 0 > U91₂ > mark₁ > ok₁
active₁ > U11₃ > mark₁ > ok₁
active₁ > U21₂ > U22₂ > mark₁ > ok₁
active₁ > U31₃ > mark₁ > ok₁
active₁ > U41₂ > U42₁ > mark₁ > ok₁
active₁ > U71₂ > U72₂ > mark₁ > ok₁
active₁ > U81₃ > U82₃ > mark₁ > ok₁
proper₁ > U102₃ > mark₁ > ok₁
proper₁ > U103₃ > mark₁ > ok₁
proper₁ > U104₃ > plus₂ > mark₁ > ok₁
proper₁ > U104₃ > x₂ > U101₃ > mark₁ > ok₁
proper₁ > U104₃ > x₂ > U61₂ > mark₁ > ok₁
proper₁ > U11₃ > mark₁ > ok₁
proper₁ > U12₃ > isNatKind₁ > U51₁ > mark₁ > ok₁
proper₁ > U12₃ > isNatKind₁ > U61₂ > mark₁ > ok₁
proper₁ > U12₃ > U13₃ > mark₁ > ok₁
proper₁ > U14₃ > U15₂ > mark₁ > ok₁
proper₁ > U16₁ > mark₁ > ok₁
proper₁ > U21₂ > U22₂ > mark₁ > ok₁
proper₁ > U23₁ > mark₁ > ok₁
proper₁ > U31₃ > mark₁ > ok₁
proper₁ > U32₃ > mark₁ > ok₁
proper₁ > U33₃ > U34₃ > U35₂ > mark₁ > ok₁
proper₁ > U41₂ > U42₁ > mark₁ > ok₁
proper₁ > U62₁ > mark₁ > ok₁
proper₁ > U71₂ > U72₂ > mark₁ > ok₁
proper₁ > U81₃ > U82₃ > mark₁ > ok₁
proper₁ > U83₃ > mark₁ > ok₁
proper₁ > U84₃ > plus₂ > mark₁ > ok₁
proper₁ > 0 > U91₂ > mark₁ > ok₁
top > ok₁

The following usable rules [FROCOS05] were oriented:

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

(187) Obligation:

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

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

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

(188) PisEmptyProof (EQUIVALENT transformation)

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

(189) TRUE

(190) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(191) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  U14¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > mark₁ > U14^1₃
active₁ > U102₃ > U103₃ > U104₃ > mark₁ > U14^1₃
active₁ > x₂ > U91₂ > mark₁ > U14^1₃
active₁ > U11₃ > U12₃ > mark₁ > U14^1₃
active₁ > U13₃ > U14₃ > U15₂ > mark₁ > U14^1₃
active₁ > U21₂ > mark₁ > U14^1₃
active₁ > U22₂ > mark₁ > U14^1₃
active₁ > U31₃ > mark₁ > U14^1₃
active₁ > U32₃ > mark₁ > U14^1₃
active₁ > U33₃ > mark₁ > U14^1₃
active₁ > U34₃ > mark₁ > U14^1₃
active₁ > U35₂ > mark₁ > U14^1₃
active₁ > U41₂ > mark₁ > U14^1₃
active₁ > U51₁ > mark₁ > U14^1₃
active₁ > U51₁ > tt > 0
active₁ > U61₂ > mark₁ > U14^1₃
active₁ > U71₂ > mark₁ > U14^1₃
active₁ > U72₂ > mark₁ > U14^1₃
active₁ > U81₃ > mark₁ > U14^1₃
active₁ > U82₃ > mark₁ > U14^1₃
active₁ > U83₃ > U84₃ > plus₂ > mark₁ > U14^1₃
proper₁ > U101₃ > mark₁ > U14^1₃
proper₁ > U102₃ > U103₃ > U104₃ > mark₁ > U14^1₃
proper₁ > x₂ > U91₂ > mark₁ > U14^1₃
proper₁ > U11₃ > U12₃ > mark₁ > U14^1₃
proper₁ > U13₃ > U14₃ > U15₂ > mark₁ > U14^1₃
proper₁ > U21₂ > mark₁ > U14^1₃
proper₁ > U22₂ > mark₁ > U14^1₃
proper₁ > U31₃ > mark₁ > U14^1₃
proper₁ > U32₃ > mark₁ > U14^1₃
proper₁ > U33₃ > mark₁ > U14^1₃
proper₁ > U34₃ > mark₁ > U14^1₃
proper₁ > U35₂ > mark₁ > U14^1₃
proper₁ > U41₂ > mark₁ > U14^1₃
proper₁ > U51₁ > mark₁ > U14^1₃
proper₁ > U51₁ > tt > 0
proper₁ > U61₂ > mark₁ > U14^1₃
proper₁ > U71₂ > mark₁ > U14^1₃
proper₁ > U72₂ > mark₁ > U14^1₃
proper₁ > U81₃ > mark₁ > U14^1₃
proper₁ > U82₃ > mark₁ > U14^1₃
proper₁ > U83₃ > U84₃ > plus₂ > mark₁ > U14^1₃

The following usable rules [FROCOS05] were oriented:

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

(192) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(193) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U14¹(x1, x2, x3)  =  U14¹(x1, x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x2, x3)
tt  =  tt
mark(x1)  =  x1
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x2, x3)
U15(x1, x2)  =  U15(x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x2
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x2, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  x2
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  x2
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > tt > U102₃ > U103₃ > ok₁
top > active₁ > tt > isNatKind₁ > U51₁ > ok₁
top > active₁ > tt > isNatKind₁ > U61₂ > ok₁
top > active₁ > tt > isNat₁ > U11₃ > ok₁
top > active₁ > tt > isNat₁ > U31₃ > U32₃ > ok₁
top > active₁ > tt > U104₂ > plus₂ > U81₃ > U82₃ > ok₁
top > active₁ > tt > x₂ > U101₂ > ok₁
top > active₁ > tt > U12₃ > U13₃ > ok₁
top > active₁ > tt > U14₂ > U15₁ > ok₁
top > active₁ > tt > U16₁ > ok₁
top > active₁ > tt > U22₁ > ok₁
top > active₁ > tt > U23₁ > ok₁
top > active₁ > tt > U33₂ > ok₁
top > active₁ > tt > U42₁ > ok₁
top > active₁ > tt > U72₂ > ok₁
top > active₁ > tt > U83₂ > U84₃ > plus₂ > U81₃ > U82₃ > ok₁
top > active₁ > tt > s₁ > U101₂ > ok₁
top > active₁ > tt > s₁ > U51₁ > ok₁
top > active₁ > tt > s₁ > U81₃ > U82₃ > ok₁
top > active₁ > tt > U92₁ > ok₁
top > active₁ > U41₂ > U42₁ > ok₁
top > active₁ > 0 > isNat₁ > U11₃ > ok₁
top > active₁ > 0 > isNat₁ > U31₃ > U32₃ > ok₁
top > active₁ > 0 > U91₂ > U92₁ > ok₁
top > proper₁ > U102₃ > U103₃ > ok₁
top > proper₁ > isNatKind₁ > U51₁ > ok₁
top > proper₁ > isNatKind₁ > U61₂ > ok₁
top > proper₁ > isNat₁ > U11₃ > ok₁
top > proper₁ > isNat₁ > U31₃ > U32₃ > ok₁
top > proper₁ > U104₂ > plus₂ > U81₃ > U82₃ > ok₁
top > proper₁ > x₂ > U101₂ > ok₁
top > proper₁ > U12₃ > U13₃ > ok₁
top > proper₁ > U14₂ > U15₁ > ok₁
top > proper₁ > U16₁ > ok₁
top > proper₁ > U22₁ > ok₁
top > proper₁ > U23₁ > ok₁
top > proper₁ > U33₂ > ok₁
top > proper₁ > U41₂ > U42₁ > ok₁
top > proper₁ > U72₂ > ok₁
top > proper₁ > U83₂ > U84₃ > plus₂ > U81₃ > U82₃ > ok₁
top > proper₁ > s₁ > U101₂ > ok₁
top > proper₁ > s₁ > U51₁ > ok₁
top > proper₁ > s₁ > U81₃ > U82₃ > ok₁
top > proper₁ > U91₂ > U92₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(194) Obligation:

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

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

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

(195) PisEmptyProof (EQUIVALENT transformation)

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

(196) TRUE

(197) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  U13¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > mark₁ > U13^1₃
active₁ > U102₃ > U103₃ > U104₃ > mark₁ > U13^1₃
active₁ > x₂ > U91₂ > mark₁ > U13^1₃
active₁ > U11₃ > U12₃ > mark₁ > U13^1₃
active₁ > U13₃ > U14₃ > U15₂ > mark₁ > U13^1₃
active₁ > U21₂ > mark₁ > U13^1₃
active₁ > U22₂ > mark₁ > U13^1₃
active₁ > U31₃ > mark₁ > U13^1₃
active₁ > U32₃ > mark₁ > U13^1₃
active₁ > U33₃ > mark₁ > U13^1₃
active₁ > U34₃ > mark₁ > U13^1₃
active₁ > U35₂ > mark₁ > U13^1₃
active₁ > U41₂ > mark₁ > U13^1₃
active₁ > U51₁ > mark₁ > U13^1₃
active₁ > U51₁ > tt > 0
active₁ > U61₂ > mark₁ > U13^1₃
active₁ > U71₂ > mark₁ > U13^1₃
active₁ > U72₂ > mark₁ > U13^1₃
active₁ > U81₃ > mark₁ > U13^1₃
active₁ > U82₃ > mark₁ > U13^1₃
active₁ > U83₃ > U84₃ > plus₂ > mark₁ > U13^1₃
proper₁ > U101₃ > mark₁ > U13^1₃
proper₁ > U102₃ > U103₃ > U104₃ > mark₁ > U13^1₃
proper₁ > x₂ > U91₂ > mark₁ > U13^1₃
proper₁ > U11₃ > U12₃ > mark₁ > U13^1₃
proper₁ > U13₃ > U14₃ > U15₂ > mark₁ > U13^1₃
proper₁ > U21₂ > mark₁ > U13^1₃
proper₁ > U22₂ > mark₁ > U13^1₃
proper₁ > U31₃ > mark₁ > U13^1₃
proper₁ > U32₃ > mark₁ > U13^1₃
proper₁ > U33₃ > mark₁ > U13^1₃
proper₁ > U34₃ > mark₁ > U13^1₃
proper₁ > U35₂ > mark₁ > U13^1₃
proper₁ > U41₂ > mark₁ > U13^1₃
proper₁ > U51₁ > mark₁ > U13^1₃
proper₁ > U51₁ > tt > 0
proper₁ > U61₂ > mark₁ > U13^1₃
proper₁ > U71₂ > mark₁ > U13^1₃
proper₁ > U72₂ > mark₁ > U13^1₃
proper₁ > U81₃ > mark₁ > U13^1₃
proper₁ > U82₃ > mark₁ > U13^1₃
proper₁ > U83₃ > U84₃ > plus₂ > mark₁ > U13^1₃

The following usable rules [FROCOS05] were oriented:

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

(199) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(200) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1, x2, x3)  =  U13¹(x1, x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x2, x3)
tt  =  tt
mark(x1)  =  x1
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x2, x3)
U15(x1, x2)  =  U15(x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x2
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x2, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  x2
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  x2
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > tt > U102₃ > U103₃ > ok₁
top > active₁ > tt > isNatKind₁ > U51₁ > ok₁
top > active₁ > tt > isNatKind₁ > U61₂ > ok₁
top > active₁ > tt > isNat₁ > U11₃ > ok₁
top > active₁ > tt > isNat₁ > U31₃ > U32₃ > ok₁
top > active₁ > tt > U104₂ > plus₂ > U81₃ > U82₃ > ok₁
top > active₁ > tt > x₂ > U101₂ > ok₁
top > active₁ > tt > U12₃ > U13₃ > ok₁
top > active₁ > tt > U14₂ > U15₁ > ok₁
top > active₁ > tt > U16₁ > ok₁
top > active₁ > tt > U22₁ > ok₁
top > active₁ > tt > U23₁ > ok₁
top > active₁ > tt > U33₂ > ok₁
top > active₁ > tt > U42₁ > ok₁
top > active₁ > tt > U72₂ > ok₁
top > active₁ > tt > U83₂ > U84₃ > plus₂ > U81₃ > U82₃ > ok₁
top > active₁ > tt > s₁ > U101₂ > ok₁
top > active₁ > tt > s₁ > U51₁ > ok₁
top > active₁ > tt > s₁ > U81₃ > U82₃ > ok₁
top > active₁ > tt > U92₁ > ok₁
top > active₁ > U41₂ > U42₁ > ok₁
top > active₁ > 0 > isNat₁ > U11₃ > ok₁
top > active₁ > 0 > isNat₁ > U31₃ > U32₃ > ok₁
top > active₁ > 0 > U91₂ > U92₁ > ok₁
top > proper₁ > U102₃ > U103₃ > ok₁
top > proper₁ > isNatKind₁ > U51₁ > ok₁
top > proper₁ > isNatKind₁ > U61₂ > ok₁
top > proper₁ > isNat₁ > U11₃ > ok₁
top > proper₁ > isNat₁ > U31₃ > U32₃ > ok₁
top > proper₁ > U104₂ > plus₂ > U81₃ > U82₃ > ok₁
top > proper₁ > x₂ > U101₂ > ok₁
top > proper₁ > U12₃ > U13₃ > ok₁
top > proper₁ > U14₂ > U15₁ > ok₁
top > proper₁ > U16₁ > ok₁
top > proper₁ > U22₁ > ok₁
top > proper₁ > U23₁ > ok₁
top > proper₁ > U33₂ > ok₁
top > proper₁ > U41₂ > U42₁ > ok₁
top > proper₁ > U72₂ > ok₁
top > proper₁ > U83₂ > U84₃ > plus₂ > U81₃ > U82₃ > ok₁
top > proper₁ > s₁ > U101₂ > ok₁
top > proper₁ > s₁ > U51₁ > ok₁
top > proper₁ > s₁ > U81₃ > U82₃ > ok₁
top > proper₁ > U91₂ > U92₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(201) Obligation:

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

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

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

(202) PisEmptyProof (EQUIVALENT transformation)

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

(203) TRUE

(204) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(205) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U104₃ > x₂ > mark₁
active₁ > plus₂ > U41₂ > mark₁
active₁ > plus₂ > U71₂ > mark₁
active₁ > U11₃ > U12₃ > mark₁
active₁ > U15₂ > mark₁
active₁ > U23₁ > tt > U102₃ > U103₃ > mark₁
active₁ > U23₁ > tt > x₂ > mark₁
active₁ > U23₁ > tt > U12₃ > mark₁
active₁ > U23₁ > tt > U13₃ > U14₃ > mark₁
active₁ > U23₁ > tt > U22₂ > mark₁
active₁ > U23₁ > tt > U32₃ > mark₁
active₁ > U23₁ > tt > U34₃ > U35₂ > mark₁
active₁ > U23₁ > tt > U83₃ > U84₃ > mark₁
active₁ > U23₁ > tt > 0
active₁ > U31₃ > mark₁
active₁ > U33₃ > U34₃ > U35₂ > mark₁
active₁ > U36₁ > tt > U102₃ > U103₃ > mark₁
active₁ > U36₁ > tt > x₂ > mark₁
active₁ > U36₁ > tt > U12₃ > mark₁
active₁ > U36₁ > tt > U13₃ > U14₃ > mark₁
active₁ > U36₁ > tt > U22₂ > mark₁
active₁ > U36₁ > tt > U32₃ > mark₁
active₁ > U36₁ > tt > U34₃ > U35₂ > mark₁
active₁ > U36₁ > tt > U83₃ > U84₃ > mark₁
active₁ > U36₁ > tt > 0
active₁ > U61₂ > mark₁
active₁ > U62₁ > tt > U102₃ > U103₃ > mark₁
active₁ > U62₁ > tt > x₂ > mark₁
active₁ > U62₁ > tt > U12₃ > mark₁
active₁ > U62₁ > tt > U13₃ > U14₃ > mark₁
active₁ > U62₁ > tt > U22₂ > mark₁
active₁ > U62₁ > tt > U32₃ > mark₁
active₁ > U62₁ > tt > U34₃ > U35₂ > mark₁
active₁ > U62₁ > tt > U83₃ > U84₃ > mark₁
active₁ > U62₁ > tt > 0
active₁ > U72₂ > mark₁
active₁ > U82₃ > U83₃ > U84₃ > mark₁
active₁ > s₁ > U101₃ > U102₃ > U103₃ > mark₁
active₁ > s₁ > U21₂ > U22₂ > mark₁
active₁ > s₁ > U51₁ > mark₁
active₁ > s₁ > U81₃ > mark₁
active₁ > U91₂ > mark₁
active₁ > U92₁ > mark₁
active₁ > U92₁ > 0

The following usable rules [FROCOS05] were oriented:

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

(206) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(207) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  x3
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  x1
U83(x1, x2, x3)  =  U83(x1, x2)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > proper₁ > U104₁ > active₁ > U101₃ > isNatKind₁ > ok₁
top > proper₁ > U104₁ > active₁ > U101₃ > isNatKind₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U103₃ > ok₁
top > proper₁ > U104₁ > active₁ > U102₃ > mark > x₂ > ok₁
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U22₁ > ok₁
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U23₁ > ok₁
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U23₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U31₃ > ok₁
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U104₁ > active₁ > U102₃ > mark > U84₃ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U103₃ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > x₂ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U22₁ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U23₁ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U23₁ > tt > 0
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U31₃ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > mark > U84₃ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > ok₁
top > proper₁ > U104₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U12₃ > isNatKind₁ > ok₁
top > proper₁ > U104₁ > active₁ > U12₃ > isNatKind₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U103₃ > ok₁
top > proper₁ > U104₁ > active₁ > U14₃ > mark > x₂ > ok₁
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U22₁ > ok₁
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U23₁ > ok₁
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U23₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U31₃ > ok₁
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U104₁ > active₁ > U14₃ > mark > U84₃ > ok₁
top > proper₁ > U104₁ > active₁ > U33₃ > isNatKind₁ > ok₁
top > proper₁ > U104₁ > active₁ > U33₃ > isNatKind₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U35₁ > ok₁
top > proper₁ > U104₁ > active₁ > U51₁ > ok₁
top > proper₁ > U104₁ > active₁ > U51₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U103₃ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > mark > x₂ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U22₁ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U23₁ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U23₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U31₃ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > mark > U84₃ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > isNatKind₁ > ok₁
top > proper₁ > U104₁ > active₁ > U83₂ > isNatKind₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U103₃ > ok₁
top > proper₁ > U104₁ > active₁ > U92₁ > mark > x₂ > ok₁
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U22₁ > ok₁
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U23₁ > ok₁
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U23₁ > tt > 0
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U31₃ > ok₁
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U104₁ > active₁ > U92₁ > mark > U84₃ > ok₁
top > proper₁ > U13₂ > active₁ > U101₃ > isNatKind₁ > ok₁
top > proper₁ > U13₂ > active₁ > U101₃ > isNatKind₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U103₃ > ok₁
top > proper₁ > U13₂ > active₁ > U102₃ > mark > x₂ > ok₁
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U22₁ > ok₁
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U23₁ > ok₁
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U23₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U31₃ > ok₁
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U13₂ > active₁ > U102₃ > mark > U84₃ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U103₃ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > x₂ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U22₁ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U23₁ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U23₁ > tt > 0
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U31₃ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > mark > U84₃ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > U81₃ > isNatKind₁ > ok₁
top > proper₁ > U13₂ > active₁ > plus₂ > U81₃ > isNatKind₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U12₃ > isNatKind₁ > ok₁
top > proper₁ > U13₂ > active₁ > U12₃ > isNatKind₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U103₃ > ok₁
top > proper₁ > U13₂ > active₁ > U14₃ > mark > x₂ > ok₁
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U22₁ > ok₁
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U23₁ > ok₁
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U23₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U31₃ > ok₁
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U13₂ > active₁ > U14₃ > mark > U84₃ > ok₁
top > proper₁ > U13₂ > active₁ > U33₃ > isNatKind₁ > ok₁
top > proper₁ > U13₂ > active₁ > U33₃ > isNatKind₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U35₁ > ok₁
top > proper₁ > U13₂ > active₁ > U51₁ > ok₁
top > proper₁ > U13₂ > active₁ > U51₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U103₃ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > mark > x₂ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U22₁ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U23₁ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U23₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U31₃ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > mark > U84₃ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > isNatKind₁ > ok₁
top > proper₁ > U13₂ > active₁ > U83₂ > isNatKind₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U103₃ > ok₁
top > proper₁ > U13₂ > active₁ > U92₁ > mark > x₂ > ok₁
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U22₁ > ok₁
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U23₁ > ok₁
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U23₁ > tt > 0
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U31₃ > ok₁
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U13₂ > active₁ > U92₁ > mark > U84₃ > ok₁
top > proper₁ > U15₁ > mark > U103₃ > ok₁
top > proper₁ > U15₁ > mark > x₂ > ok₁
top > proper₁ > U15₁ > mark > U22₁ > ok₁
top > proper₁ > U15₁ > mark > U23₁ > ok₁
top > proper₁ > U15₁ > mark > U23₁ > tt > 0
top > proper₁ > U15₁ > mark > U31₃ > ok₁
top > proper₁ > U15₁ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U15₁ > mark > U84₃ > ok₁
top > proper₁ > U21₁ > active₁ > U101₃ > isNatKind₁ > ok₁
top > proper₁ > U21₁ > active₁ > U101₃ > isNatKind₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U103₃ > ok₁
top > proper₁ > U21₁ > active₁ > U102₃ > mark > x₂ > ok₁
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U22₁ > ok₁
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U23₁ > ok₁
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U23₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U31₃ > ok₁
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U21₁ > active₁ > U102₃ > mark > U84₃ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U103₃ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > x₂ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U22₁ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U23₁ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U23₁ > tt > 0
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U31₃ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > mark > U84₃ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > ok₁
top > proper₁ > U21₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U12₃ > isNatKind₁ > ok₁
top > proper₁ > U21₁ > active₁ > U12₃ > isNatKind₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U103₃ > ok₁
top > proper₁ > U21₁ > active₁ > U14₃ > mark > x₂ > ok₁
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U22₁ > ok₁
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U23₁ > ok₁
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U23₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U31₃ > ok₁
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U21₁ > active₁ > U14₃ > mark > U84₃ > ok₁
top > proper₁ > U21₁ > active₁ > U33₃ > isNatKind₁ > ok₁
top > proper₁ > U21₁ > active₁ > U33₃ > isNatKind₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U35₁ > ok₁
top > proper₁ > U21₁ > active₁ > U51₁ > ok₁
top > proper₁ > U21₁ > active₁ > U51₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U103₃ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > mark > x₂ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U22₁ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U23₁ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U23₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U31₃ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > mark > U84₃ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > isNatKind₁ > ok₁
top > proper₁ > U21₁ > active₁ > U83₂ > isNatKind₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U103₃ > ok₁
top > proper₁ > U21₁ > active₁ > U92₁ > mark > x₂ > ok₁
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U22₁ > ok₁
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U23₁ > ok₁
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U23₁ > tt > 0
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U31₃ > ok₁
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U21₁ > active₁ > U92₁ > mark > U84₃ > ok₁
top > proper₁ > U32₁ > active₁ > U101₃ > isNatKind₁ > ok₁
top > proper₁ > U32₁ > active₁ > U101₃ > isNatKind₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U103₃ > ok₁
top > proper₁ > U32₁ > active₁ > U102₃ > mark > x₂ > ok₁
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U22₁ > ok₁
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U23₁ > ok₁
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U23₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U31₃ > ok₁
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U32₁ > active₁ > U102₃ > mark > U84₃ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U103₃ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > x₂ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U22₁ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U23₁ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U23₁ > tt > 0
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U31₃ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > mark > U84₃ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > ok₁
top > proper₁ > U32₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U12₃ > isNatKind₁ > ok₁
top > proper₁ > U32₁ > active₁ > U12₃ > isNatKind₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U103₃ > ok₁
top > proper₁ > U32₁ > active₁ > U14₃ > mark > x₂ > ok₁
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U22₁ > ok₁
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U23₁ > ok₁
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U23₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U31₃ > ok₁
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U32₁ > active₁ > U14₃ > mark > U84₃ > ok₁
top > proper₁ > U32₁ > active₁ > U33₃ > isNatKind₁ > ok₁
top > proper₁ > U32₁ > active₁ > U33₃ > isNatKind₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U35₁ > ok₁
top > proper₁ > U32₁ > active₁ > U51₁ > ok₁
top > proper₁ > U32₁ > active₁ > U51₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U103₃ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > mark > x₂ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U22₁ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U23₁ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U23₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U31₃ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > mark > U84₃ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > isNatKind₁ > ok₁
top > proper₁ > U32₁ > active₁ > U83₂ > isNatKind₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U103₃ > ok₁
top > proper₁ > U32₁ > active₁ > U92₁ > mark > x₂ > ok₁
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U22₁ > ok₁
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U23₁ > ok₁
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U23₁ > tt > 0
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U31₃ > ok₁
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U32₁ > active₁ > U92₁ > mark > U84₃ > ok₁
top > proper₁ > U91₁ > active₁ > U101₃ > isNatKind₁ > ok₁
top > proper₁ > U91₁ > active₁ > U101₃ > isNatKind₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U103₃ > ok₁
top > proper₁ > U91₁ > active₁ > U102₃ > mark > x₂ > ok₁
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U22₁ > ok₁
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U23₁ > ok₁
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U23₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U31₃ > ok₁
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U91₁ > active₁ > U102₃ > mark > U84₃ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U103₃ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > x₂ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U22₁ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U23₁ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U23₁ > tt > 0
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U31₃ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > mark > U84₃ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > ok₁
top > proper₁ > U91₁ > active₁ > plus₂ > U81₃ > isNatKind₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U12₃ > isNatKind₁ > ok₁
top > proper₁ > U91₁ > active₁ > U12₃ > isNatKind₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U103₃ > ok₁
top > proper₁ > U91₁ > active₁ > U14₃ > mark > x₂ > ok₁
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U22₁ > ok₁
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U23₁ > ok₁
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U23₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U31₃ > ok₁
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U91₁ > active₁ > U14₃ > mark > U84₃ > ok₁
top > proper₁ > U91₁ > active₁ > U33₃ > isNatKind₁ > ok₁
top > proper₁ > U91₁ > active₁ > U33₃ > isNatKind₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U35₁ > ok₁
top > proper₁ > U91₁ > active₁ > U51₁ > ok₁
top > proper₁ > U91₁ > active₁ > U51₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U103₃ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > mark > x₂ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U22₁ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U23₁ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U23₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U31₃ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > mark > U84₃ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > isNatKind₁ > ok₁
top > proper₁ > U91₁ > active₁ > U83₂ > isNatKind₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U103₃ > ok₁
top > proper₁ > U91₁ > active₁ > U92₁ > mark > x₂ > ok₁
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U22₁ > ok₁
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U23₁ > ok₁
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U23₁ > tt > 0
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U31₃ > ok₁
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U71₂ > U72₁ > ok₁
top > proper₁ > U91₁ > active₁ > U92₁ > mark > U84₃ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(208) Obligation:

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

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

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

(209) PisEmptyProof (EQUIVALENT transformation)

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

(210) TRUE

(211) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(212) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > mark₁
active₁ > U11₃ > U12₃ > mark₁
active₁ > U31₃ > mark₁
active₁ > U41₂ > U42₁ > mark₁
active₁ > U81₃ > mark₁
active₁ > 0 > tt > U102₃ > mark₁
active₁ > 0 > tt > U103₃ > U104₃ > plus₂ > mark₁
active₁ > 0 > tt > isNat₁ > U21₂ > mark₁
active₁ > 0 > tt > x₂ > U61₂ > mark₁
active₁ > 0 > tt > U12₃ > mark₁
active₁ > 0 > tt > U13₃ > U14₃ > mark₁
active₁ > 0 > tt > U15₂ > mark₁
active₁ > 0 > tt > U22₂ > U23₁ > mark₁
active₁ > 0 > tt > U32₃ > mark₁
active₁ > 0 > tt > U33₃ > mark₁
active₁ > 0 > tt > U34₃ > mark₁
active₁ > 0 > tt > U35₂ > mark₁
active₁ > 0 > tt > U42₁ > mark₁
active₁ > 0 > tt > U72₂ > mark₁
active₁ > 0 > tt > U82₃ > U83₃ > mark₁
active₁ > 0 > tt > U84₃ > s₁ > mark₁
active₁ > 0 > U71₂ > mark₁
active₁ > 0 > U91₂ > mark₁
proper₁ > U101₃ > mark₁
proper₁ > U102₃ > mark₁
proper₁ > U103₃ > U104₃ > plus₂ > mark₁
proper₁ > isNat₁ > U21₂ > mark₁
proper₁ > x₂ > U61₂ > mark₁
proper₁ > U11₃ > U12₃ > mark₁
proper₁ > U13₃ > U14₃ > mark₁
proper₁ > U15₂ > mark₁
proper₁ > U22₂ > U23₁ > mark₁
proper₁ > U31₃ > mark₁
proper₁ > U32₃ > mark₁
proper₁ > U33₃ > mark₁
proper₁ > U34₃ > mark₁
proper₁ > U35₂ > mark₁
proper₁ > U41₂ > U42₁ > mark₁
proper₁ > U71₂ > mark₁
proper₁ > U72₂ > mark₁
proper₁ > U81₃ > mark₁
proper₁ > U82₃ > U83₃ > mark₁
proper₁ > U84₃ > s₁ > mark₁
proper₁ > U91₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(213) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(214) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2, x3)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1)
U72(x1, x2)  =  x1
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  x1
U83(x1, x2, x3)  =  U83(x1)
U84(x1, x2, x3)  =  U84(x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

0 > isNat₁ > tt > plus₂ > ok₁
0 > isNat₁ > tt > plus₂ > mark
0 > isNat₁ > tt > x₁ > U31₃ > ok₁
0 > isNat₁ > tt > x₁ > U31₃ > mark
0 > isNat₁ > tt > U16₁ > ok₁
0 > isNat₁ > tt > U16₁ > mark
0 > isNat₁ > tt > U33₃ > ok₁
0 > isNat₁ > tt > U33₃ > mark
0 > isNat₁ > tt > U42₁ > ok₁
0 > isNat₁ > tt > U42₁ > mark
0 > isNat₁ > tt > s₁ > ok₁
0 > isNat₁ > tt > s₁ > mark
0 > isNat₁ > U11₂ > ok₁
0 > isNat₁ > U11₂ > mark
0 > U71₁ > ok₁
0 > U71₁ > mark
0 > U91₂ > ok₁
0 > U91₂ > mark
proper₁ > U101₃ > ok₁
proper₁ > U101₃ > mark
proper₁ > U102₃ > ok₁
proper₁ > U102₃ > mark
proper₁ > U103₃ > U104₃ > plus₂ > ok₁
proper₁ > U103₃ > U104₃ > plus₂ > mark
proper₁ > U103₃ > U104₃ > x₁ > U31₃ > ok₁
proper₁ > U103₃ > U104₃ > x₁ > U31₃ > mark
proper₁ > U13₃ > ok₁
proper₁ > U13₃ > mark
proper₁ > U15₂ > isNat₁ > tt > plus₂ > ok₁
proper₁ > U15₂ > isNat₁ > tt > plus₂ > mark
proper₁ > U15₂ > isNat₁ > tt > x₁ > U31₃ > ok₁
proper₁ > U15₂ > isNat₁ > tt > x₁ > U31₃ > mark
proper₁ > U15₂ > isNat₁ > tt > U16₁ > ok₁
proper₁ > U15₂ > isNat₁ > tt > U16₁ > mark
proper₁ > U15₂ > isNat₁ > tt > U33₃ > ok₁
proper₁ > U15₂ > isNat₁ > tt > U33₃ > mark
proper₁ > U15₂ > isNat₁ > tt > U42₁ > ok₁
proper₁ > U15₂ > isNat₁ > tt > U42₁ > mark
proper₁ > U15₂ > isNat₁ > tt > s₁ > ok₁
proper₁ > U15₂ > isNat₁ > tt > s₁ > mark
proper₁ > U15₂ > isNat₁ > U11₂ > ok₁
proper₁ > U15₂ > isNat₁ > U11₂ > mark
proper₁ > U21₁ > ok₁
proper₁ > U21₁ > mark
proper₁ > U22₁ > ok₁
proper₁ > U22₁ > mark
proper₁ > U32₃ > U33₃ > ok₁
proper₁ > U32₃ > U33₃ > mark
proper₁ > U35₁ > ok₁
proper₁ > U35₁ > mark
proper₁ > U41₁ > U42₁ > ok₁
proper₁ > U41₁ > U42₁ > mark
proper₁ > U51₁ > tt > plus₂ > ok₁
proper₁ > U51₁ > tt > plus₂ > mark
proper₁ > U51₁ > tt > x₁ > U31₃ > ok₁
proper₁ > U51₁ > tt > x₁ > U31₃ > mark
proper₁ > U51₁ > tt > U16₁ > ok₁
proper₁ > U51₁ > tt > U16₁ > mark
proper₁ > U51₁ > tt > U33₃ > ok₁
proper₁ > U51₁ > tt > U33₃ > mark
proper₁ > U51₁ > tt > U42₁ > ok₁
proper₁ > U51₁ > tt > U42₁ > mark
proper₁ > U51₁ > tt > s₁ > ok₁
proper₁ > U51₁ > tt > s₁ > mark
proper₁ > U61₂ > ok₁
proper₁ > U61₂ > mark
proper₁ > U71₁ > ok₁
proper₁ > U71₁ > mark
proper₁ > U81₃ > ok₁
proper₁ > U81₃ > mark
proper₁ > U83₁ > U84₂ > plus₂ > ok₁
proper₁ > U83₁ > U84₂ > plus₂ > mark
proper₁ > U83₁ > U84₂ > s₁ > ok₁
proper₁ > U83₁ > U84₂ > s₁ > mark
proper₁ > U91₂ > ok₁
proper₁ > U91₂ > mark

The following usable rules [FROCOS05] were oriented:

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

(215) Obligation:

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

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

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

(216) PisEmptyProof (EQUIVALENT transformation)

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

(217) TRUE

(218) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(219) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  X(x1, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

X₂ > isNat
active₁ > U103₃ > mark₁ > isNat
active₁ > U104₃ > plus₂ > U71₂ > U72₂ > mark₁ > isNat
active₁ > U104₃ > x₂ > U101₃ > U102₃ > mark₁ > isNat
active₁ > U16₁ > mark₁ > isNat
active₁ > U16₁ > tt > isNat
active₁ > U21₁ > mark₁ > isNat
active₁ > U31₁ > mark₁ > isNat
active₁ > U32₁ > U33₁ > mark₁ > isNat
active₁ > U61₁ > isNatKind > mark₁ > isNat
active₁ > U61₁ > isNatKind > tt > isNat
active₁ > U81₃ > isNatKind > mark₁ > isNat
active₁ > U81₃ > isNatKind > tt > isNat
active₁ > U81₃ > U82₃ > mark₁ > isNat
active₁ > U83₃ > mark₁ > isNat
active₁ > U84₃ > mark₁ > isNat
active₁ > U92₁ > mark₁ > isNat
active₁ > U92₁ > 0 > tt > isNat
top > isNat

The following usable rules [FROCOS05] were oriented:

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

(220) Obligation:

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

X(ok(X1), ok(X2)) → X(X1, X2)

The TRS R consists of the following rules:

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

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

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

X(ok(X1), ok(X2)) → X(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  x2
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
active₁ > U11₃ > mark₁ > ok₁
active₁ > U21₂ > U22₂ > mark₁ > ok₁
active₁ > U31₃ > mark₁ > ok₁
active₁ > U32₃ > U33₃ > mark₁ > ok₁
active₁ > U41₂ > mark₁ > ok₁
active₁ > U51₁ > tt > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
active₁ > U51₁ > tt > x₂ > mark₁ > ok₁
active₁ > U51₁ > tt > U12₃ > U13₃ > U14₃ > isNat₁ > mark₁ > ok₁
active₁ > U51₁ > tt > U12₃ > U13₃ > U14₃ > U15₂ > mark₁ > ok₁
active₁ > U51₁ > tt > U22₂ > mark₁ > ok₁
active₁ > U51₁ > tt > U33₃ > mark₁ > ok₁
active₁ > U51₁ > tt > U34₃ > mark₁ > ok₁
active₁ > U51₁ > tt > U35₂ > U36₁ > mark₁ > ok₁
active₁ > U51₁ > tt > U82₃ > U83₃ > mark₁ > ok₁
active₁ > U51₁ > tt > U84₃ > plus₂ > U71₂ > mark₁ > ok₁
active₁ > U51₁ > tt > U84₃ > s₁ > mark₁ > ok₁
active₁ > U51₁ > tt > U92₁ > mark₁ > ok₁
active₁ > U51₁ > tt > U92₁ > 0 > ok₁
active₁ > U61₂ > mark₁ > ok₁
active₁ > U62₁ > tt > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
active₁ > U62₁ > tt > x₂ > mark₁ > ok₁
active₁ > U62₁ > tt > U12₃ > U13₃ > U14₃ > isNat₁ > mark₁ > ok₁
active₁ > U62₁ > tt > U12₃ > U13₃ > U14₃ > U15₂ > mark₁ > ok₁
active₁ > U62₁ > tt > U22₂ > mark₁ > ok₁
active₁ > U62₁ > tt > U33₃ > mark₁ > ok₁
active₁ > U62₁ > tt > U34₃ > mark₁ > ok₁
active₁ > U62₁ > tt > U35₂ > U36₁ > mark₁ > ok₁
active₁ > U62₁ > tt > U82₃ > U83₃ > mark₁ > ok₁
active₁ > U62₁ > tt > U84₃ > plus₂ > U71₂ > mark₁ > ok₁
active₁ > U62₁ > tt > U84₃ > s₁ > mark₁ > ok₁
active₁ > U62₁ > tt > U92₁ > mark₁ > ok₁
active₁ > U62₁ > tt > U92₁ > 0 > ok₁
active₁ > U72₂ > mark₁ > ok₁
active₁ > U81₃ > mark₁ > ok₁
active₁ > U91₂ > U92₁ > mark₁ > ok₁
active₁ > U91₂ > U92₁ > 0 > ok₁
proper₁ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
proper₁ > U11₃ > mark₁ > ok₁
proper₁ > U21₂ > U22₂ > mark₁ > ok₁
proper₁ > U31₃ > mark₁ > ok₁
proper₁ > U32₃ > U33₃ > mark₁ > ok₁
proper₁ > U41₂ > mark₁ > ok₁
proper₁ > U51₁ > tt > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
proper₁ > U51₁ > tt > x₂ > mark₁ > ok₁
proper₁ > U51₁ > tt > U12₃ > U13₃ > U14₃ > isNat₁ > mark₁ > ok₁
proper₁ > U51₁ > tt > U12₃ > U13₃ > U14₃ > U15₂ > mark₁ > ok₁
proper₁ > U51₁ > tt > U22₂ > mark₁ > ok₁
proper₁ > U51₁ > tt > U33₃ > mark₁ > ok₁
proper₁ > U51₁ > tt > U34₃ > mark₁ > ok₁
proper₁ > U51₁ > tt > U35₂ > U36₁ > mark₁ > ok₁
proper₁ > U51₁ > tt > U82₃ > U83₃ > mark₁ > ok₁
proper₁ > U51₁ > tt > U84₃ > plus₂ > U71₂ > mark₁ > ok₁
proper₁ > U51₁ > tt > U84₃ > s₁ > mark₁ > ok₁
proper₁ > U51₁ > tt > U92₁ > mark₁ > ok₁
proper₁ > U51₁ > tt > U92₁ > 0 > ok₁
proper₁ > U61₂ > mark₁ > ok₁
proper₁ > U62₁ > tt > U102₃ > U103₃ > U104₃ > mark₁ > ok₁
proper₁ > U62₁ > tt > x₂ > mark₁ > ok₁
proper₁ > U62₁ > tt > U12₃ > U13₃ > U14₃ > isNat₁ > mark₁ > ok₁
proper₁ > U62₁ > tt > U12₃ > U13₃ > U14₃ > U15₂ > mark₁ > ok₁
proper₁ > U62₁ > tt > U22₂ > mark₁ > ok₁
proper₁ > U62₁ > tt > U33₃ > mark₁ > ok₁
proper₁ > U62₁ > tt > U34₃ > mark₁ > ok₁
proper₁ > U62₁ > tt > U35₂ > U36₁ > mark₁ > ok₁
proper₁ > U62₁ > tt > U82₃ > U83₃ > mark₁ > ok₁
proper₁ > U62₁ > tt > U84₃ > plus₂ > U71₂ > mark₁ > ok₁
proper₁ > U62₁ > tt > U84₃ > s₁ > mark₁ > ok₁
proper₁ > U62₁ > tt > U92₁ > mark₁ > ok₁
proper₁ > U62₁ > tt > U92₁ > 0 > ok₁
proper₁ > U72₂ > mark₁ > ok₁
proper₁ > U81₃ > mark₁ > ok₁
proper₁ > U91₂ > U92₁ > mark₁ > ok₁
proper₁ > U91₂ > U92₁ > 0 > ok₁
top > ok₁

The following usable rules [FROCOS05] were oriented:

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

(222) Obligation:

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

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

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

(223) PisEmptyProof (EQUIVALENT transformation)

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

(224) TRUE

(225) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(226) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

PLUS₂ > isNat
active₁ > U101₃ > U102₃ > mark₁ > isNat
active₁ > isNatKind > tt > isNat
active₁ > isNatKind > U61₁ > mark₁ > isNat
active₁ > U103₃ > mark₁ > isNat
active₁ > U104₃ > plus₂ > mark₁ > isNat
active₁ > U104₃ > x₂ > U91₂ > mark₁ > isNat
active₁ > U16₁ > mark₁ > isNat
active₁ > U16₁ > tt > isNat
active₁ > U21₁ > mark₁ > isNat
active₁ > U31₁ > U32₁ > U33₁ > mark₁ > isNat
active₁ > U71₂ > U72₂ > mark₁ > isNat
active₁ > U81₃ > U82₃ > mark₁ > isNat
active₁ > U83₃ > U84₃ > plus₂ > mark₁ > isNat
active₁ > 0 > tt > isNat
top > isNat

The following usable rules [FROCOS05] were oriented:

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

(227) Obligation:

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

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

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

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

(228) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > tt > U103₃ > ok₁
active₁ > tt > U103₃ > mark₁
active₁ > tt > U104₃ > x₂ > U101₃ > U102₃ > ok₁
active₁ > tt > U104₃ > x₂ > U101₃ > U102₃ > mark₁
active₁ > tt > plus₂ > U11₃ > ok₁
active₁ > tt > plus₂ > U11₃ > mark₁
active₁ > tt > plus₂ > U71₂ > U72₂ > ok₁
active₁ > tt > plus₂ > U71₂ > U72₂ > mark₁
active₁ > tt > plus₂ > U81₃ > ok₁
active₁ > tt > plus₂ > U81₃ > mark₁
active₁ > tt > U12₃ > U13₃ > U14₃ > U15₂ > ok₁
active₁ > tt > U12₃ > U13₃ > U14₃ > U15₂ > mark₁
active₁ > tt > U16₁ > ok₁
active₁ > tt > U16₁ > mark₁
active₁ > tt > U22₂ > ok₁
active₁ > tt > U22₂ > mark₁
active₁ > tt > U32₃ > U33₃ > ok₁
active₁ > tt > U32₃ > U33₃ > mark₁
active₁ > tt > U34₃ > U35₂ > ok₁
active₁ > tt > U34₃ > U35₂ > mark₁
active₁ > tt > U62₁ > ok₁
active₁ > tt > U62₁ > mark₁
active₁ > tt > U82₃ > U83₃ > ok₁
active₁ > tt > U82₃ > U83₃ > mark₁
active₁ > tt > U84₃ > ok₁
active₁ > tt > U84₃ > mark₁
active₁ > tt > U92₁ > ok₁
active₁ > tt > U92₁ > mark₁
active₁ > U21₂ > U22₂ > ok₁
active₁ > U21₂ > U22₂ > mark₁
active₁ > U31₃ > U32₃ > U33₃ > ok₁
active₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U41₂ > ok₁
active₁ > U41₂ > mark₁
active₁ > U51₁ > ok₁
active₁ > U51₁ > mark₁
active₁ > U61₂ > U62₁ > ok₁
active₁ > U61₂ > U62₁ > mark₁
active₁ > 0 > U71₂ > U72₂ > ok₁
active₁ > 0 > U71₂ > U72₂ > mark₁
active₁ > 0 > U91₂ > ok₁
active₁ > 0 > U91₂ > mark₁
proper₁ > U103₃ > ok₁
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > x₂ > U101₃ > U102₃ > ok₁
proper₁ > U104₃ > x₂ > U101₃ > U102₃ > mark₁
proper₁ > plus₂ > U11₃ > ok₁
proper₁ > plus₂ > U11₃ > mark₁
proper₁ > plus₂ > U71₂ > U72₂ > ok₁
proper₁ > plus₂ > U71₂ > U72₂ > mark₁
proper₁ > plus₂ > U81₃ > ok₁
proper₁ > plus₂ > U81₃ > mark₁
proper₁ > U12₃ > U13₃ > U14₃ > U15₂ > ok₁
proper₁ > U12₃ > U13₃ > U14₃ > U15₂ > mark₁
proper₁ > U16₁ > ok₁
proper₁ > U16₁ > mark₁
proper₁ > U21₂ > U22₂ > ok₁
proper₁ > U21₂ > U22₂ > mark₁
proper₁ > U31₃ > U32₃ > U33₃ > ok₁
proper₁ > U31₃ > U32₃ > U33₃ > mark₁
proper₁ > U34₃ > U35₂ > ok₁
proper₁ > U34₃ > U35₂ > mark₁
proper₁ > U41₂ > ok₁
proper₁ > U41₂ > mark₁
proper₁ > U51₁ > ok₁
proper₁ > U51₁ > mark₁
proper₁ > U61₂ > U62₁ > ok₁
proper₁ > U61₂ > U62₁ > mark₁
proper₁ > U82₃ > U83₃ > ok₁
proper₁ > U82₃ > U83₃ > mark₁
proper₁ > U84₃ > ok₁
proper₁ > U84₃ > mark₁
proper₁ > U91₂ > ok₁
proper₁ > U91₂ > mark₁
proper₁ > U92₁ > ok₁
proper₁ > U92₁ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(229) Obligation:

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

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

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

(230) PisEmptyProof (EQUIVALENT transformation)

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

(231) TRUE

(232) Obligation:

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

U104¹(ok(X1), ok(X2), ok(X3)) → U104¹(X1, X2, X3)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(233) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  U104¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > mark₁ > U104^1₃
active₁ > U102₃ > U103₃ > U104₃ > mark₁ > U104^1₃
active₁ > x₂ > U91₂ > mark₁ > U104^1₃
active₁ > U11₃ > U12₃ > mark₁ > U104^1₃
active₁ > U13₃ > U14₃ > U15₂ > mark₁ > U104^1₃
active₁ > U21₂ > mark₁ > U104^1₃
active₁ > U22₂ > mark₁ > U104^1₃
active₁ > U31₃ > mark₁ > U104^1₃
active₁ > U32₃ > mark₁ > U104^1₃
active₁ > U33₃ > mark₁ > U104^1₃
active₁ > U34₃ > mark₁ > U104^1₃
active₁ > U35₂ > mark₁ > U104^1₃
active₁ > U41₂ > mark₁ > U104^1₃
active₁ > U51₁ > mark₁ > U104^1₃
active₁ > U51₁ > tt > 0
active₁ > U61₂ > mark₁ > U104^1₃
active₁ > U71₂ > mark₁ > U104^1₃
active₁ > U72₂ > mark₁ > U104^1₃
active₁ > U81₃ > mark₁ > U104^1₃
active₁ > U82₃ > mark₁ > U104^1₃
active₁ > U83₃ > U84₃ > plus₂ > mark₁ > U104^1₃
proper₁ > U101₃ > mark₁ > U104^1₃
proper₁ > U102₃ > U103₃ > U104₃ > mark₁ > U104^1₃
proper₁ > x₂ > U91₂ > mark₁ > U104^1₃
proper₁ > U11₃ > U12₃ > mark₁ > U104^1₃
proper₁ > U13₃ > U14₃ > U15₂ > mark₁ > U104^1₃
proper₁ > U21₂ > mark₁ > U104^1₃
proper₁ > U22₂ > mark₁ > U104^1₃
proper₁ > U31₃ > mark₁ > U104^1₃
proper₁ > U32₃ > mark₁ > U104^1₃
proper₁ > U33₃ > mark₁ > U104^1₃
proper₁ > U34₃ > mark₁ > U104^1₃
proper₁ > U35₂ > mark₁ > U104^1₃
proper₁ > U41₂ > mark₁ > U104^1₃
proper₁ > U51₁ > mark₁ > U104^1₃
proper₁ > U51₁ > tt > 0
proper₁ > U61₂ > mark₁ > U104^1₃
proper₁ > U71₂ > mark₁ > U104^1₃
proper₁ > U72₂ > mark₁ > U104^1₃
proper₁ > U81₃ > mark₁ > U104^1₃
proper₁ > U82₃ > mark₁ > U104^1₃
proper₁ > U83₃ > U84₃ > plus₂ > mark₁ > U104^1₃

The following usable rules [FROCOS05] were oriented:

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

(234) Obligation:

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

U104¹(ok(X1), ok(X2), ok(X3)) → U104¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(235) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(ok(X1), ok(X2), ok(X3)) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  U104¹(x1, x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x2, x3)
tt  =  tt
mark(x1)  =  x1
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x2, x3)
U15(x1, x2)  =  U15(x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x2
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x2, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  x2
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  x2
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > tt > U102₃ > U103₃ > ok₁
top > active₁ > tt > isNatKind₁ > U51₁ > ok₁
top > active₁ > tt > isNatKind₁ > U61₂ > ok₁
top > active₁ > tt > isNat₁ > U11₃ > ok₁
top > active₁ > tt > isNat₁ > U31₃ > U32₃ > ok₁
top > active₁ > tt > U104₂ > plus₂ > U81₃ > U82₃ > ok₁
top > active₁ > tt > x₂ > U101₂ > ok₁
top > active₁ > tt > U12₃ > U13₃ > ok₁
top > active₁ > tt > U14₂ > U15₁ > ok₁
top > active₁ > tt > U16₁ > ok₁
top > active₁ > tt > U22₁ > ok₁
top > active₁ > tt > U23₁ > ok₁
top > active₁ > tt > U33₂ > ok₁
top > active₁ > tt > U42₁ > ok₁
top > active₁ > tt > U72₂ > ok₁
top > active₁ > tt > U83₂ > U84₃ > plus₂ > U81₃ > U82₃ > ok₁
top > active₁ > tt > s₁ > U101₂ > ok₁
top > active₁ > tt > s₁ > U51₁ > ok₁
top > active₁ > tt > s₁ > U81₃ > U82₃ > ok₁
top > active₁ > tt > U92₁ > ok₁
top > active₁ > U41₂ > U42₁ > ok₁
top > active₁ > 0 > isNat₁ > U11₃ > ok₁
top > active₁ > 0 > isNat₁ > U31₃ > U32₃ > ok₁
top > active₁ > 0 > U91₂ > U92₁ > ok₁
top > proper₁ > U102₃ > U103₃ > ok₁
top > proper₁ > isNatKind₁ > U51₁ > ok₁
top > proper₁ > isNatKind₁ > U61₂ > ok₁
top > proper₁ > isNat₁ > U11₃ > ok₁
top > proper₁ > isNat₁ > U31₃ > U32₃ > ok₁
top > proper₁ > U104₂ > plus₂ > U81₃ > U82₃ > ok₁
top > proper₁ > x₂ > U101₂ > ok₁
top > proper₁ > U12₃ > U13₃ > ok₁
top > proper₁ > U14₂ > U15₁ > ok₁
top > proper₁ > U16₁ > ok₁
top > proper₁ > U22₁ > ok₁
top > proper₁ > U23₁ > ok₁
top > proper₁ > U33₂ > ok₁
top > proper₁ > U41₂ > U42₁ > ok₁
top > proper₁ > U72₂ > ok₁
top > proper₁ > U83₂ > U84₃ > plus₂ > U81₃ > U82₃ > ok₁
top > proper₁ > s₁ > U101₂ > ok₁
top > proper₁ > s₁ > U51₁ > ok₁
top > proper₁ > s₁ > U81₃ > U82₃ > ok₁
top > proper₁ > U91₂ > U92₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(236) Obligation:

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

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

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

(237) PisEmptyProof (EQUIVALENT transformation)

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

(238) TRUE

(239) Obligation:

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

U103¹(ok(X1), ok(X2), ok(X3)) → U103¹(X1, X2, X3)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(240) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(ok(X1), ok(X2), ok(X3)) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  U103¹(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x3
U12(x1, x2, x3)  =  x3
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x3
U32(x1, x2, x3)  =  U32(x3)
U33(x1, x2, x3)  =  U33(x1, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  U35(x2)
U36(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x2
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x2)
U72(x1, x2)  =  x2
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U103^1₃ > ok₁
active₁ > U101₃ > ok₁
active₁ > tt > U102₃ > U103₃ > ok₁
active₁ > tt > U102₃ > isNat₁ > U21₁ > ok₁
active₁ > tt > U104₂ > ok₁
active₁ > tt > plus₂ > U71₁ > ok₁
active₁ > tt > plus₂ > U81₃ > ok₁
active₁ > tt > x₂ > ok₁
active₁ > tt > U23₁ > ok₁
active₁ > tt > U32₁ > U33₂ > ok₁
active₁ > tt > U35₁ > ok₁
active₁ > tt > U42₁ > ok₁
active₁ > tt > U62₁ > ok₁
active₁ > tt > U82₃ > U83₃ > U84₃ > ok₁
active₁ > tt > U92₁ > ok₁
active₁ > U51₁ > ok₁
active₁ > 0 > isNat₁ > U21₁ > ok₁
active₁ > 0 > U71₁ > ok₁
active₁ > 0 > U91₂ > ok₁
top > proper₁ > U101₃ > ok₁
top > proper₁ > U102₃ > U103₃ > ok₁
top > proper₁ > U102₃ > isNat₁ > U21₁ > ok₁
top > proper₁ > U104₂ > ok₁
top > proper₁ > plus₂ > U71₁ > ok₁
top > proper₁ > plus₂ > U81₃ > ok₁
top > proper₁ > x₂ > ok₁
top > proper₁ > U23₁ > ok₁
top > proper₁ > U32₁ > U33₂ > ok₁
top > proper₁ > U35₁ > ok₁
top > proper₁ > U42₁ > ok₁
top > proper₁ > U51₁ > ok₁
top > proper₁ > U62₁ > ok₁
top > proper₁ > U82₃ > U83₃ > U84₃ > ok₁
top > proper₁ > U91₂ > ok₁
top > proper₁ > U92₁ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(241) Obligation:

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

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(242) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > isNat > proper₁ > U101₃ > mark₁
top > active₁ > isNat > proper₁ > U101₃ > ok
top > active₁ > isNat > proper₁ > U102₃ > mark₁
top > active₁ > isNat > proper₁ > U102₃ > ok
top > active₁ > isNat > proper₁ > isNatKind > mark₁
top > active₁ > isNat > proper₁ > isNatKind > tt
top > active₁ > isNat > proper₁ > isNatKind > ok
top > active₁ > isNat > proper₁ > U103₃ > mark₁
top > active₁ > isNat > proper₁ > U103₃ > ok
top > active₁ > isNat > proper₁ > U104₃ > mark₁
top > active₁ > isNat > proper₁ > U104₃ > ok
top > active₁ > isNat > proper₁ > plus₂ > mark₁
top > active₁ > isNat > proper₁ > plus₂ > ok
top > active₁ > isNat > proper₁ > x₂ > mark₁
top > active₁ > isNat > proper₁ > x₂ > ok
top > active₁ > isNat > proper₁ > U11₁ > mark₁
top > active₁ > isNat > proper₁ > U11₁ > ok
top > active₁ > isNat > proper₁ > U12₁ > mark₁
top > active₁ > isNat > proper₁ > U12₁ > ok
top > active₁ > isNat > proper₁ > U15₁ > mark₁
top > active₁ > isNat > proper₁ > U15₁ > ok
top > active₁ > isNat > proper₁ > U21₁ > mark₁
top > active₁ > isNat > proper₁ > U21₁ > ok
top > active₁ > isNat > proper₁ > U22₁ > mark₁
top > active₁ > isNat > proper₁ > U22₁ > ok
top > active₁ > isNat > proper₁ > U23₁ > mark₁
top > active₁ > isNat > proper₁ > U23₁ > tt
top > active₁ > isNat > proper₁ > U23₁ > ok
top > active₁ > isNat > proper₁ > U33₁ > mark₁
top > active₁ > isNat > proper₁ > U33₁ > ok
top > active₁ > isNat > proper₁ > U34₁ > mark₁
top > active₁ > isNat > proper₁ > U34₁ > ok
top > active₁ > isNat > proper₁ > U41₁ > mark₁
top > active₁ > isNat > proper₁ > U41₁ > ok
top > active₁ > isNat > proper₁ > U61₁ > mark₁
top > active₁ > isNat > proper₁ > U61₁ > ok
top > active₁ > isNat > proper₁ > U71₂ > mark₁
top > active₁ > isNat > proper₁ > U71₂ > ok
top > active₁ > isNat > proper₁ > U72₂ > mark₁
top > active₁ > isNat > proper₁ > U72₂ > ok
top > active₁ > isNat > proper₁ > U81₃ > mark₁
top > active₁ > isNat > proper₁ > U81₃ > ok
top > active₁ > isNat > proper₁ > U82₃ > mark₁
top > active₁ > isNat > proper₁ > U82₃ > ok
top > active₁ > isNat > proper₁ > U83₃ > mark₁
top > active₁ > isNat > proper₁ > U83₃ > ok
top > active₁ > isNat > proper₁ > U84₃ > mark₁
top > active₁ > isNat > proper₁ > U84₃ > ok
top > active₁ > isNat > proper₁ > U91₂ > mark₁
top > active₁ > isNat > proper₁ > U91₂ > ok
top > active₁ > 0 > tt
top > active₁ > 0 > U71₂ > mark₁
top > active₁ > 0 > U71₂ > ok
top > active₁ > 0 > U91₂ > mark₁
top > active₁ > 0 > U91₂ > ok

The following usable rules [FROCOS05] were oriented:

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

(243) Obligation:

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

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

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

(244) PisEmptyProof (EQUIVALENT transformation)

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

(245) TRUE

(246) Obligation:

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

U102¹(ok(X1), ok(X2), ok(X3)) → U102¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(247) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  U102¹(x1, x2, x3)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

U102^1₃ > mark₁
active₁ > U101₃ > U102₃ > mark₁
active₁ > isNatKind > U41₁ > U42₁ > mark₁
active₁ > isNatKind > U51₁ > tt > mark₁
active₁ > isNatKind > U61₁ > mark₁
active₁ > U103₃ > mark₁
active₁ > U104₃ > plus₂ > U11₃ > mark₁
active₁ > U104₃ > plus₂ > U41₁ > U42₁ > mark₁
active₁ > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > U12₃ > U13₃ > mark₁
active₁ > U14₃ > U15₂ > mark₁
active₁ > U16₁ > tt > mark₁
active₁ > U21₂ > U22₂ > mark₁
active₁ > U23₁ > tt > mark₁
active₁ > U31₃ > U32₃ > mark₁
active₁ > U33₃ > U34₃ > mark₁
active₁ > U35₂ > mark₁
active₁ > U36₁ > tt > mark₁
active₁ > U62₁ > tt > mark₁
active₁ > U82₃ > mark₁
active₁ > U83₃ > U84₃ > s₁ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U71₂ > U72₂ > mark₁
active₁ > 0 > U91₂ > U92₁ > mark₁
proper₁ > U101₃ > U102₃ > mark₁
proper₁ > U103₃ > mark₁
proper₁ > U104₃ > plus₂ > U11₃ > mark₁
proper₁ > U104₃ > plus₂ > U41₁ > U42₁ > mark₁
proper₁ > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U104₃ > x₂ > mark₁
proper₁ > U12₃ > U13₃ > mark₁
proper₁ > U14₃ > U15₂ > mark₁
proper₁ > U16₁ > tt > mark₁
proper₁ > U21₂ > U22₂ > mark₁
proper₁ > U23₁ > tt > mark₁
proper₁ > U31₃ > U32₃ > mark₁
proper₁ > U33₃ > U34₃ > mark₁
proper₁ > U35₂ > mark₁
proper₁ > U36₁ > tt > mark₁
proper₁ > U51₁ > tt > mark₁
proper₁ > U61₁ > mark₁
proper₁ > U62₁ > tt > mark₁
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U82₃ > mark₁
proper₁ > U83₃ > U84₃ > s₁ > mark₁
proper₁ > U91₂ > U92₁ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(248) Obligation:

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

U102¹(ok(X1), ok(X2), ok(X3)) → U102¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(249) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(ok(X1), ok(X2), ok(X3)) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  U102¹(x1, x2)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x2)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  x3
plus(x1, x2)  =  x2
x(x1, x2)  =  x2
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x2
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x2)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x2
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x2, x3)
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  x2
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x2)
U72(x1, x2)  =  x1
U81(x1, x2, x3)  =  x2
U82(x1, x2, x3)  =  x3
U83(x1, x2, x3)  =  x2
U84(x1, x2, x3)  =  x1
s(x1)  =  x1
U91(x1, x2)  =  U91(x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

proper₁ > U101₁ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > U101₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U103₂ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > U103₂ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U11₂ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > U11₂ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U15₁ > ok₁ > U102^1₂ > mark
proper₁ > U15₁ > ok₁ > active₁ > tt > mark
proper₁ > U31₂ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > U31₂ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U35₁ > ok₁ > U102^1₂ > mark
proper₁ > U35₁ > ok₁ > active₁ > tt > mark
proper₁ > U41₁ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > U41₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > U71₁ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > U71₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
proper₁ > 0 > U91₁ > isNatKind₁ > ok₁ > U102^1₂ > mark
proper₁ > 0 > U91₁ > isNatKind₁ > ok₁ > active₁ > tt > mark
top > active₁ > tt > mark

The following usable rules [FROCOS05] were oriented:

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

(250) Obligation:

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

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

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

(251) PisEmptyProof (EQUIVALENT transformation)

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

(252) TRUE

(253) Obligation:

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

U101¹(ok(X1), ok(X2), ok(X3)) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(254) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(ok(X1), ok(X2), ok(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x3
U12(x1, x2, x3)  =  U12(x3)
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x3
U32(x1, x2, x3)  =  x3
U33(x1, x2, x3)  =  U33(x1, x3)
U34(x1, x2, x3)  =  x3
U35(x1, x2)  =  U35(x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x2)
U42(x1)  =  U42(x1)
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x2
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x2)
U72(x1, x2)  =  x2
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > ok₁
active₁ > U101₃ > isNatKind₁ > U41₁ > ok₁
active₁ > U101₃ > isNatKind₁ > U51₁ > ok₁
active₁ > tt > U102₃ > ok₁
active₁ > tt > isNatKind₁ > U41₁ > ok₁
active₁ > tt > isNatKind₁ > U51₁ > ok₁
active₁ > tt > U104₃ > x₂ > ok₁
active₁ > tt > U23₁ > ok₁
active₁ > tt > U33₂ > ok₁
active₁ > tt > U35₁ > ok₁
active₁ > tt > U36₁ > ok₁
active₁ > tt > U42₁ > ok₁
active₁ > tt > U62₁ > ok₁
active₁ > tt > U82₃ > U83₂ > ok₁
active₁ > tt > s₁ > U51₁ > ok₁
active₁ > U103₃ > isNatKind₁ > U41₁ > ok₁
active₁ > U103₃ > isNatKind₁ > U51₁ > ok₁
active₁ > U103₃ > U104₃ > x₂ > ok₁
active₁ > plus₂ > isNatKind₁ > U41₁ > ok₁
active₁ > plus₂ > isNatKind₁ > U51₁ > ok₁
active₁ > plus₂ > U81₃ > U82₃ > U83₂ > ok₁
active₁ > U12₁ > isNatKind₁ > U41₁ > ok₁
active₁ > U12₁ > isNatKind₁ > U51₁ > ok₁
active₁ > U71₁ > isNatKind₁ > U41₁ > ok₁
active₁ > U71₁ > isNatKind₁ > U51₁ > ok₁
active₁ > U84₂ > s₁ > U51₁ > ok₁
active₁ > 0 > U91₂ > ok₁
proper₁ > U101₃ > U102₃ > ok₁
proper₁ > U101₃ > isNatKind₁ > U41₁ > ok₁
proper₁ > U101₃ > isNatKind₁ > U51₁ > ok₁
proper₁ > tt > U102₃ > ok₁
proper₁ > tt > isNatKind₁ > U41₁ > ok₁
proper₁ > tt > isNatKind₁ > U51₁ > ok₁
proper₁ > tt > U104₃ > x₂ > ok₁
proper₁ > tt > U23₁ > ok₁
proper₁ > tt > U33₂ > ok₁
proper₁ > tt > U35₁ > ok₁
proper₁ > tt > U36₁ > ok₁
proper₁ > tt > U42₁ > ok₁
proper₁ > tt > U62₁ > ok₁
proper₁ > tt > U82₃ > U83₂ > ok₁
proper₁ > tt > s₁ > U51₁ > ok₁
proper₁ > U103₃ > isNatKind₁ > U41₁ > ok₁
proper₁ > U103₃ > isNatKind₁ > U51₁ > ok₁
proper₁ > U103₃ > U104₃ > x₂ > ok₁
proper₁ > plus₂ > isNatKind₁ > U41₁ > ok₁
proper₁ > plus₂ > isNatKind₁ > U51₁ > ok₁
proper₁ > plus₂ > U81₃ > U82₃ > U83₂ > ok₁
proper₁ > U12₁ > isNatKind₁ > U41₁ > ok₁
proper₁ > U12₁ > isNatKind₁ > U51₁ > ok₁
proper₁ > U71₁ > isNatKind₁ > U41₁ > ok₁
proper₁ > U71₁ > isNatKind₁ > U51₁ > ok₁
proper₁ > U84₂ > s₁ > U51₁ > ok₁
proper₁ > 0 > U91₂ > ok₁

The following usable rules [FROCOS05] were oriented:

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

(255) Obligation:

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

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The TRS R consists of the following rules:

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

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

(256) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U102₃ > U103₃ > mark₁
active₁ > U102₃ > isNat
active₁ > isNatKind > U41₁ > mark₁
active₁ > isNatKind > U51₁ > mark₁
active₁ > isNatKind > U51₁ > tt
active₁ > U104₃ > x₂ > U101₃ > mark₁
active₁ > U104₃ > x₂ > isNat
active₁ > plus₂ > U41₁ > mark₁
active₁ > plus₂ > U71₂ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > mark₁
active₁ > plus₂ > U81₃ > U82₃ > isNat
active₁ > U13₁ > mark₁
active₁ > U14₁ > mark₁
active₁ > U14₁ > isNat
active₁ > U22₁ > mark₁
active₁ > U22₁ > isNat
active₁ > U23₁ > mark₁
active₁ > U23₁ > tt
active₁ > U33₁ > mark₁
active₁ > U72₂ > mark₁
active₁ > U83₃ > mark₁
active₁ > U84₃ > mark₁
active₁ > s₁ > U101₃ > mark₁
active₁ > s₁ > U51₁ > mark₁
active₁ > s₁ > U51₁ > tt
active₁ > s₁ > U81₃ > U82₃ > mark₁
active₁ > s₁ > U81₃ > U82₃ > isNat
active₁ > U92₁ > mark₁
active₁ > 0 > tt
active₁ > 0 > isNat
active₁ > 0 > U71₂ > mark₁
proper₁ > U102₃ > U103₃ > mark₁
proper₁ > U102₃ > isNat
proper₁ > isNatKind > U41₁ > mark₁
proper₁ > isNatKind > U51₁ > mark₁
proper₁ > isNatKind > U51₁ > tt
proper₁ > U104₃ > x₂ > U101₃ > mark₁
proper₁ > U104₃ > x₂ > isNat
proper₁ > plus₂ > U41₁ > mark₁
proper₁ > plus₂ > U71₂ > mark₁
proper₁ > plus₂ > U81₃ > U82₃ > mark₁
proper₁ > plus₂ > U81₃ > U82₃ > isNat
proper₁ > U13₁ > mark₁
proper₁ > U14₁ > mark₁
proper₁ > U14₁ > isNat
proper₁ > U22₁ > mark₁
proper₁ > U22₁ > isNat
proper₁ > U23₁ > mark₁
proper₁ > U23₁ > tt
proper₁ > U33₁ > mark₁
proper₁ > U72₂ > mark₁
proper₁ > U83₃ > mark₁
proper₁ > U84₃ > mark₁
proper₁ > s₁ > U101₃ > mark₁
proper₁ > s₁ > U51₁ > mark₁
proper₁ > s₁ > U51₁ > tt
proper₁ > s₁ > U81₃ > U82₃ > mark₁
proper₁ > s₁ > U81₃ > U82₃ > isNat
proper₁ > U92₁ > mark₁
proper₁ > 0 > tt
proper₁ > 0 > isNat
proper₁ > 0 > U71₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(257) Obligation:

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

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

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

(258) PisEmptyProof (EQUIVALENT transformation)

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

(259) TRUE

(260) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(261) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U101(X1, X2, X3)) → PROPER(X2)
PROPER(U101(X1, X2, X3)) → PROPER(X1)
PROPER(U101(X1, X2, X3)) → PROPER(X3)
PROPER(U102(X1, X2, X3)) → PROPER(X1)
PROPER(U102(X1, X2, X3)) → PROPER(X2)
PROPER(U102(X1, X2, X3)) → PROPER(X3)
PROPER(U103(X1, X2, X3)) → PROPER(X1)
PROPER(U103(X1, X2, X3)) → PROPER(X2)
PROPER(U103(X1, X2, X3)) → PROPER(X3)
PROPER(U104(X1, X2, X3)) → PROPER(X1)
PROPER(U104(X1, X2, X3)) → PROPER(X2)
PROPER(U104(X1, X2, X3)) → PROPER(X3)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X1, X2, X3)) → PROPER(X2)
PROPER(U12(X1, X2, X3)) → PROPER(X3)
PROPER(U13(X1, X2, X3)) → PROPER(X1)
PROPER(U13(X1, X2, X3)) → PROPER(X2)
PROPER(U13(X1, X2, X3)) → PROPER(X3)
PROPER(U14(X1, X2, X3)) → PROPER(X1)
PROPER(U14(X1, X2, X3)) → PROPER(X2)
PROPER(U14(X1, X2, X3)) → PROPER(X3)
PROPER(U15(X1, X2)) → PROPER(X1)
PROPER(U15(X1, X2)) → PROPER(X2)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X1, X2)) → PROPER(X1)
PROPER(U22(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2, X3)) → PROPER(X1)
PROPER(U31(X1, X2, X3)) → PROPER(X2)
PROPER(U31(X1, X2, X3)) → PROPER(X3)
PROPER(U32(X1, X2, X3)) → PROPER(X1)
PROPER(U32(X1, X2, X3)) → PROPER(X2)
PROPER(U32(X1, X2, X3)) → PROPER(X3)
PROPER(U33(X1, X2, X3)) → PROPER(X1)
PROPER(U33(X1, X2, X3)) → PROPER(X2)
PROPER(U33(X1, X2, X3)) → PROPER(X3)
PROPER(U34(X1, X2, X3)) → PROPER(X1)
PROPER(U34(X1, X2, X3)) → PROPER(X2)
PROPER(U34(X1, X2, X3)) → PROPER(X3)
PROPER(U35(X1, X2)) → PROPER(X1)
PROPER(U35(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U72(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2)) → PROPER(X2)
PROPER(U81(X1, X2, X3)) → PROPER(X1)
PROPER(U81(X1, X2, X3)) → PROPER(X2)
PROPER(U81(X1, X2, X3)) → PROPER(X3)
PROPER(U82(X1, X2, X3)) → PROPER(X1)
PROPER(U82(X1, X2, X3)) → PROPER(X2)
PROPER(U82(X1, X2, X3)) → PROPER(X3)
PROPER(U83(X1, X2, X3)) → PROPER(X1)
PROPER(U83(X1, X2, X3)) → PROPER(X2)
PROPER(U83(X1, X2, X3)) → PROPER(X3)
PROPER(U84(X1, X2, X3)) → PROPER(X1)
PROPER(U84(X1, X2, X3)) → PROPER(X2)
PROPER(U84(X1, X2, X3)) → PROPER(X3)
PROPER(U91(X1, X2)) → PROPER(X1)
PROPER(U91(X1, X2)) → PROPER(X2)
PROPER(U92(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  mark
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > mark > proper₁ > U101₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U101₃ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > U102₃ > U103₃ > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U102₃ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > x₂ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U11₃ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > PROPER₁ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > U15₂ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U13₃ > U14₃ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > U21₂ > PROPER₁ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U21₂ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U31₃ > ok > U92₁ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U12₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U22₂ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U32₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U33₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U34₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U35₂ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U71₂ > U72₂ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U82₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U83₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U84₃ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U91₂ > U81₃
active₁ > mark > proper₁ > U61₂ > ok > U92₁ > U81₃
active₁ > mark > top > U81₃
tt > mark > proper₁ > U101₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U101₃ > ok > U12₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U22₂ > U81₃
tt > mark > proper₁ > U101₃ > ok > U32₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U33₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U34₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U35₂ > U81₃
tt > mark > proper₁ > U101₃ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U101₃ > ok > U82₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U83₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U84₃ > U81₃
tt > mark > proper₁ > U101₃ > ok > U91₂ > U81₃
tt > mark > proper₁ > U101₃ > ok > U92₁ > U81₃
tt > mark > proper₁ > U102₃ > U103₃ > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U102₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U102₃ > ok > U12₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U22₂ > U81₃
tt > mark > proper₁ > U102₃ > ok > U32₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U33₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U34₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U35₂ > U81₃
tt > mark > proper₁ > U102₃ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U102₃ > ok > U82₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U83₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U84₃ > U81₃
tt > mark > proper₁ > U102₃ > ok > U91₂ > U81₃
tt > mark > proper₁ > U102₃ > ok > U92₁ > U81₃
tt > mark > proper₁ > x₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > x₂ > ok > U12₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U22₂ > U81₃
tt > mark > proper₁ > x₂ > ok > U32₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U33₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U34₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U35₂ > U81₃
tt > mark > proper₁ > x₂ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > x₂ > ok > U82₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U83₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U84₃ > U81₃
tt > mark > proper₁ > x₂ > ok > U91₂ > U81₃
tt > mark > proper₁ > x₂ > ok > U92₁ > U81₃
tt > mark > proper₁ > U11₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U11₃ > ok > U12₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U22₂ > U81₃
tt > mark > proper₁ > U11₃ > ok > U32₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U33₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U34₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U35₂ > U81₃
tt > mark > proper₁ > U11₃ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U11₃ > ok > U82₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U83₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U84₃ > U81₃
tt > mark > proper₁ > U11₃ > ok > U91₂ > U81₃
tt > mark > proper₁ > U11₃ > ok > U92₁ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > PROPER₁ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > U15₂ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U12₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U22₂ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U32₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U33₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U34₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U35₂ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U82₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U83₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U84₃ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U91₂ > U81₃
tt > mark > proper₁ > U13₃ > U14₃ > ok > U92₁ > U81₃
tt > mark > proper₁ > U21₂ > PROPER₁ > U81₃
tt > mark > proper₁ > U21₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U21₂ > ok > U12₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U22₂ > U81₃
tt > mark > proper₁ > U21₂ > ok > U32₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U33₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U34₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U35₂ > U81₃
tt > mark > proper₁ > U21₂ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U21₂ > ok > U82₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U83₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U84₃ > U81₃
tt > mark > proper₁ > U21₂ > ok > U91₂ > U81₃
tt > mark > proper₁ > U21₂ > ok > U92₁ > U81₃
tt > mark > proper₁ > U31₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U31₃ > ok > U12₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U22₂ > U81₃
tt > mark > proper₁ > U31₃ > ok > U32₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U33₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U34₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U35₂ > U81₃
tt > mark > proper₁ > U31₃ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U31₃ > ok > U82₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U83₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U84₃ > U81₃
tt > mark > proper₁ > U31₃ > ok > U91₂ > U81₃
tt > mark > proper₁ > U31₃ > ok > U92₁ > U81₃
tt > mark > proper₁ > U61₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
tt > mark > proper₁ > U61₂ > ok > U12₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U22₂ > U81₃
tt > mark > proper₁ > U61₂ > ok > U32₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U33₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U34₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U35₂ > U81₃
tt > mark > proper₁ > U61₂ > ok > U71₂ > U72₂ > U81₃
tt > mark > proper₁ > U61₂ > ok > U82₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U83₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U84₃ > U81₃
tt > mark > proper₁ > U61₂ > ok > U91₂ > U81₃
tt > mark > proper₁ > U61₂ > ok > U92₁ > U81₃
tt > mark > top > U81₃
0 > mark > proper₁ > U101₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U101₃ > ok > U12₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U22₂ > U81₃
0 > mark > proper₁ > U101₃ > ok > U32₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U33₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U34₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U35₂ > U81₃
0 > mark > proper₁ > U101₃ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U101₃ > ok > U82₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U83₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U84₃ > U81₃
0 > mark > proper₁ > U101₃ > ok > U91₂ > U81₃
0 > mark > proper₁ > U101₃ > ok > U92₁ > U81₃
0 > mark > proper₁ > U102₃ > U103₃ > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U102₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U102₃ > ok > U12₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U22₂ > U81₃
0 > mark > proper₁ > U102₃ > ok > U32₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U33₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U34₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U35₂ > U81₃
0 > mark > proper₁ > U102₃ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U102₃ > ok > U82₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U83₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U84₃ > U81₃
0 > mark > proper₁ > U102₃ > ok > U91₂ > U81₃
0 > mark > proper₁ > U102₃ > ok > U92₁ > U81₃
0 > mark > proper₁ > x₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > x₂ > ok > U12₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U22₂ > U81₃
0 > mark > proper₁ > x₂ > ok > U32₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U33₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U34₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U35₂ > U81₃
0 > mark > proper₁ > x₂ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > x₂ > ok > U82₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U83₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U84₃ > U81₃
0 > mark > proper₁ > x₂ > ok > U91₂ > U81₃
0 > mark > proper₁ > x₂ > ok > U92₁ > U81₃
0 > mark > proper₁ > U11₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U11₃ > ok > U12₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U22₂ > U81₃
0 > mark > proper₁ > U11₃ > ok > U32₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U33₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U34₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U35₂ > U81₃
0 > mark > proper₁ > U11₃ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U11₃ > ok > U82₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U83₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U84₃ > U81₃
0 > mark > proper₁ > U11₃ > ok > U91₂ > U81₃
0 > mark > proper₁ > U11₃ > ok > U92₁ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > PROPER₁ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > U15₂ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U12₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U22₂ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U32₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U33₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U34₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U35₂ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U82₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U83₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U84₃ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U91₂ > U81₃
0 > mark > proper₁ > U13₃ > U14₃ > ok > U92₁ > U81₃
0 > mark > proper₁ > U21₂ > PROPER₁ > U81₃
0 > mark > proper₁ > U21₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U21₂ > ok > U12₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U22₂ > U81₃
0 > mark > proper₁ > U21₂ > ok > U32₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U33₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U34₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U35₂ > U81₃
0 > mark > proper₁ > U21₂ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U21₂ > ok > U82₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U83₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U84₃ > U81₃
0 > mark > proper₁ > U21₂ > ok > U91₂ > U81₃
0 > mark > proper₁ > U21₂ > ok > U92₁ > U81₃
0 > mark > proper₁ > U31₃ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U31₃ > ok > U12₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U22₂ > U81₃
0 > mark > proper₁ > U31₃ > ok > U32₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U33₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U34₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U35₂ > U81₃
0 > mark > proper₁ > U31₃ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U31₃ > ok > U82₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U83₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U84₃ > U81₃
0 > mark > proper₁ > U31₃ > ok > U91₂ > U81₃
0 > mark > proper₁ > U31₃ > ok > U92₁ > U81₃
0 > mark > proper₁ > U61₂ > ok > U104₃ > plus₂ > U41₂ > U81₃
0 > mark > proper₁ > U61₂ > ok > U12₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U22₂ > U81₃
0 > mark > proper₁ > U61₂ > ok > U32₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U33₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U34₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U35₂ > U81₃
0 > mark > proper₁ > U61₂ > ok > U71₂ > U72₂ > U81₃
0 > mark > proper₁ > U61₂ > ok > U82₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U83₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U84₃ > U81₃
0 > mark > proper₁ > U61₂ > ok > U91₂ > U81₃
0 > mark > proper₁ > U61₂ > ok > U92₁ > U81₃
0 > mark > top > U81₃

The following usable rules [FROCOS05] were oriented:

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

(262) Obligation:

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

PROPER(isNatKind(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U16(X)) → PROPER(X)
PROPER(U23(X)) → PROPER(X)
PROPER(U36(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U51(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(263) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U42(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatKind(x1)  =  x1
isNat(x1)  =  x1
U16(x1)  =  x1
U23(x1)  =  x1
U36(x1)  =  x1
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U62(x1)  =  x1
s(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark
U102(x1, x2, x3)  =  U102(x1)
U103(x1, x2, x3)  =  U103(x2, x3)
U104(x1, x2, x3)  =  x1
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x3)
U13(x1, x2, x3)  =  U13(x1, x2)
U14(x1, x2, x3)  =  U14(x1, x2)
U15(x1, x2)  =  U15
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x2)
U33(x1, x2, x3)  =  U33(x2)
U34(x1, x2, x3)  =  U34(x2)
U35(x1, x2)  =  U35(x1)
U41(x1, x2)  =  U41(x1, x2)
U61(x1, x2)  =  U61(x1, x2)
U71(x1, x2)  =  x1
U72(x1, x2)  =  U72(x2)
U81(x1, x2, x3)  =  U81(x2, x3)
U82(x1, x2, x3)  =  U82(x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

tt > 0 > ok > U103₂ > mark > plus₂
tt > 0 > ok > U103₂ > mark > U41₂
tt > 0 > ok > U103₂ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > U42₁ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > U42₁ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > U42₁ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > U102₁ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > U102₁ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > U102₁ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > U11₃
tt > 0 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > U14₂ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > U14₂ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > U14₂ > mark > U92₁
tt > 0 > ok > U34₁ > active₁ > U35₁ > mark > plus₂
tt > 0 > ok > U34₁ > active₁ > U35₁ > mark > U41₂
tt > 0 > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U15 > ok > U103₂ > mark > plus₂
U15 > ok > U103₂ > mark > U41₂
U15 > ok > U103₂ > mark > U92₁
U15 > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U15 > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U15 > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U15 > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U15 > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U15 > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U15 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U15 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U15 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U15 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U15 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U15 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U15 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U15 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U15 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U15 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U15 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U15 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U15 > ok > U34₁ > active₁ > U11₃
U15 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U15 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U15 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U15 > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U15 > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U15 > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U15 > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U15 > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U15 > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U22 > ok > U103₂ > mark > plus₂
U22 > ok > U103₂ > mark > U41₂
U22 > ok > U103₂ > mark > U92₁
U22 > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U22 > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U22 > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U22 > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U22 > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U22 > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U22 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U22 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U22 > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U22 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U22 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U22 > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U22 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U22 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U22 > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U22 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U22 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U22 > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U22 > ok > U34₁ > active₁ > U11₃
U22 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U22 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U22 > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U22 > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U22 > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U22 > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U22 > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U22 > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U22 > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U32₁ > ok > U103₂ > mark > plus₂
U32₁ > ok > U103₂ > mark > U41₂
U32₁ > ok > U103₂ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > U11₃
U32₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U32₁ > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U32₁ > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U32₁ > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U33₁ > ok > U103₂ > mark > plus₂
U33₁ > ok > U103₂ > mark > U41₂
U33₁ > ok > U103₂ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > U11₃
U33₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U33₁ > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U33₁ > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U33₁ > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U72₁ > ok > U103₂ > mark > plus₂
U72₁ > ok > U103₂ > mark > U41₂
U72₁ > ok > U103₂ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > U11₃
U72₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U72₁ > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U72₁ > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U72₁ > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U81₂ > ok > U103₂ > mark > plus₂
U81₂ > ok > U103₂ > mark > U41₂
U81₂ > ok > U103₂ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > U11₃
U81₂ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U81₂ > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U81₂ > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U81₂ > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U82₁ > ok > U103₂ > mark > plus₂
U82₁ > ok > U103₂ > mark > U41₂
U82₁ > ok > U103₂ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > U11₃
U82₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U82₁ > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U82₁ > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U82₁ > ok > U34₁ > active₁ > U35₁ > mark > U92₁
U83₂ > ok > U103₂ > mark > plus₂
U83₂ > ok > U103₂ > mark > U41₂
U83₂ > ok > U103₂ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > U42₁ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > U42₁ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > U42₁ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > U102₁ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > U102₁ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > U102₁ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > x₁ > U101₃ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > x₁ > U31₃ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > x₁ > U61₂ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > x₁ > U91₂ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > U11₃
U83₂ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > U12₂ > U13₂ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > U14₂ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > U14₂ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > U14₂ > mark > U92₁
U83₂ > ok > U34₁ > active₁ > U35₁ > mark > plus₂
U83₂ > ok > U34₁ > active₁ > U35₁ > mark > U41₂
U83₂ > ok > U34₁ > active₁ > U35₁ > mark > U92₁
top > active₁ > U42₁ > mark > plus₂
top > active₁ > U42₁ > mark > U41₂
top > active₁ > U42₁ > mark > U92₁
top > active₁ > U102₁ > mark > plus₂
top > active₁ > U102₁ > mark > U41₂
top > active₁ > U102₁ > mark > U92₁
top > active₁ > x₁ > U101₃ > mark > plus₂
top > active₁ > x₁ > U101₃ > mark > U41₂
top > active₁ > x₁ > U101₃ > mark > U92₁
top > active₁ > x₁ > U31₃ > mark > plus₂
top > active₁ > x₁ > U31₃ > mark > U41₂
top > active₁ > x₁ > U31₃ > mark > U92₁
top > active₁ > x₁ > U61₂ > mark > plus₂
top > active₁ > x₁ > U61₂ > mark > U41₂
top > active₁ > x₁ > U61₂ > mark > U92₁
top > active₁ > x₁ > U91₂ > mark > plus₂
top > active₁ > x₁ > U91₂ > mark > U41₂
top > active₁ > x₁ > U91₂ > mark > U92₁
top > active₁ > U11₃
top > active₁ > U12₂ > U13₂ > mark > plus₂
top > active₁ > U12₂ > U13₂ > mark > U41₂
top > active₁ > U12₂ > U13₂ > mark > U92₁
top > active₁ > U14₂ > mark > plus₂
top > active₁ > U14₂ > mark > U41₂
top > active₁ > U14₂ > mark > U92₁
top > active₁ > U35₁ > mark > plus₂
top > active₁ > U35₁ > mark > U41₂
top > active₁ > U35₁ > mark > U92₁

The following usable rules [FROCOS05] were oriented:

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

(264) Obligation:

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

PROPER(isNatKind(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U16(X)) → PROPER(X)
PROPER(U23(X)) → PROPER(X)
PROPER(U36(X)) → PROPER(X)
PROPER(U51(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(265) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNatKind(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U16(X)) → PROPER(X)
PROPER(U36(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatKind(x1)  =  isNatKind(x1)
isNat(x1)  =  isNat(x1)
U16(x1)  =  U16(x1)
U23(x1)  =  x1
U36(x1)  =  U36(x1)
U51(x1)  =  x1
U62(x1)  =  x1
s(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U61(x1, x2)  =  U61(x1, x2)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U36₁ > tt > U16₁ > mark₁
active₁ > U36₁ > tt > U16₁ > ok₁
active₁ > U36₁ > tt > plus₂ > isNatKind₁ > ok₁
active₁ > U36₁ > tt > plus₂ > isNat₁ > ok₁
active₁ > U36₁ > tt > plus₂ > mark₁
active₁ > U36₁ > tt > x₂ > isNatKind₁ > ok₁
active₁ > U36₁ > tt > x₂ > isNat₁ > ok₁
active₁ > U36₁ > tt > x₂ > mark₁
active₁ > U36₁ > tt > U13₃ > isNatKind₁ > ok₁
active₁ > U36₁ > tt > U13₃ > mark₁
active₁ > U36₁ > tt > U22₂ > isNat₁ > ok₁
active₁ > U36₁ > tt > U22₂ > mark₁
active₁ > U36₁ > tt > U32₃ > isNatKind₁ > ok₁
active₁ > U36₁ > tt > U32₃ > mark₁
active₁ > U36₁ > tt > U33₃ > isNatKind₁ > ok₁
active₁ > U36₁ > tt > U33₃ > mark₁
active₁ > U36₁ > tt > U35₂ > isNat₁ > ok₁
active₁ > U36₁ > tt > U35₂ > mark₁
active₁ > U101₃ > isNatKind₁ > ok₁
active₁ > U101₃ > mark₁
active₁ > U102₃ > U103₃ > U104₃ > plus₂ > isNatKind₁ > ok₁
active₁ > U102₃ > U103₃ > U104₃ > plus₂ > isNat₁ > ok₁
active₁ > U102₃ > U103₃ > U104₃ > plus₂ > mark₁
active₁ > U102₃ > U103₃ > U104₃ > x₂ > isNatKind₁ > ok₁
active₁ > U102₃ > U103₃ > U104₃ > x₂ > isNat₁ > ok₁
active₁ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
active₁ > U11₃ > isNatKind₁ > ok₁
active₁ > U11₃ > mark₁
active₁ > U12₃ > U13₃ > isNatKind₁ > ok₁
active₁ > U12₃ > U13₃ > mark₁
active₁ > U14₃ > isNat₁ > ok₁
active₁ > U14₃ > mark₁
active₁ > U15₂ > isNat₁ > ok₁
active₁ > U15₂ > U16₁ > mark₁
active₁ > U15₂ > U16₁ > ok₁
active₁ > U21₂ > isNatKind₁ > ok₁
active₁ > U21₂ > mark₁
active₁ > U31₃ > isNatKind₁ > ok₁
active₁ > U31₃ > mark₁
active₁ > U34₃ > U35₂ > isNat₁ > ok₁
active₁ > U34₃ > U35₂ > mark₁
active₁ > U41₂ > isNatKind₁ > ok₁
active₁ > U41₂ > mark₁
active₁ > U42₁ > tt > U16₁ > mark₁
active₁ > U42₁ > tt > U16₁ > ok₁
active₁ > U42₁ > tt > plus₂ > isNatKind₁ > ok₁
active₁ > U42₁ > tt > plus₂ > isNat₁ > ok₁
active₁ > U42₁ > tt > plus₂ > mark₁
active₁ > U42₁ > tt > x₂ > isNatKind₁ > ok₁
active₁ > U42₁ > tt > x₂ > isNat₁ > ok₁
active₁ > U42₁ > tt > x₂ > mark₁
active₁ > U42₁ > tt > U13₃ > isNatKind₁ > ok₁
active₁ > U42₁ > tt > U13₃ > mark₁
active₁ > U42₁ > tt > U22₂ > isNat₁ > ok₁
active₁ > U42₁ > tt > U22₂ > mark₁
active₁ > U42₁ > tt > U32₃ > isNatKind₁ > ok₁
active₁ > U42₁ > tt > U32₃ > mark₁
active₁ > U42₁ > tt > U33₃ > isNatKind₁ > ok₁
active₁ > U42₁ > tt > U33₃ > mark₁
active₁ > U42₁ > tt > U35₂ > isNat₁ > ok₁
active₁ > U42₁ > tt > U35₂ > mark₁
active₁ > U61₂ > isNatKind₁ > ok₁
active₁ > U61₂ > mark₁
active₁ > U71₂ > isNatKind₁ > ok₁
active₁ > U71₂ > U72₂ > mark₁
active₁ > U71₂ > U72₂ > ok₁
active₁ > U81₃ > isNatKind₁ > ok₁
active₁ > U81₃ > U82₃ > isNat₁ > ok₁
active₁ > U81₃ > U82₃ > mark₁
active₁ > U83₃ > isNatKind₁ > ok₁
active₁ > U83₃ > mark₁
active₁ > U84₃ > plus₂ > isNatKind₁ > ok₁
active₁ > U84₃ > plus₂ > isNat₁ > ok₁
active₁ > U84₃ > plus₂ > mark₁
active₁ > 0 > isNat₁ > ok₁
active₁ > 0 > U91₂ > isNatKind₁ > ok₁
active₁ > 0 > U91₂ > mark₁
proper₁ > U36₁ > tt > U16₁ > mark₁
proper₁ > U36₁ > tt > U16₁ > ok₁
proper₁ > U36₁ > tt > plus₂ > isNatKind₁ > ok₁
proper₁ > U36₁ > tt > plus₂ > isNat₁ > ok₁
proper₁ > U36₁ > tt > plus₂ > mark₁
proper₁ > U36₁ > tt > x₂ > isNatKind₁ > ok₁
proper₁ > U36₁ > tt > x₂ > isNat₁ > ok₁
proper₁ > U36₁ > tt > x₂ > mark₁
proper₁ > U36₁ > tt > U13₃ > isNatKind₁ > ok₁
proper₁ > U36₁ > tt > U13₃ > mark₁
proper₁ > U36₁ > tt > U22₂ > isNat₁ > ok₁
proper₁ > U36₁ > tt > U22₂ > mark₁
proper₁ > U36₁ > tt > U32₃ > isNatKind₁ > ok₁
proper₁ > U36₁ > tt > U32₃ > mark₁
proper₁ > U36₁ > tt > U33₃ > isNatKind₁ > ok₁
proper₁ > U36₁ > tt > U33₃ > mark₁
proper₁ > U36₁ > tt > U35₂ > isNat₁ > ok₁
proper₁ > U36₁ > tt > U35₂ > mark₁
proper₁ > U101₃ > isNatKind₁ > ok₁
proper₁ > U101₃ > mark₁
proper₁ > U102₃ > U103₃ > U104₃ > plus₂ > isNatKind₁ > ok₁
proper₁ > U102₃ > U103₃ > U104₃ > plus₂ > isNat₁ > ok₁
proper₁ > U102₃ > U103₃ > U104₃ > plus₂ > mark₁
proper₁ > U102₃ > U103₃ > U104₃ > x₂ > isNatKind₁ > ok₁
proper₁ > U102₃ > U103₃ > U104₃ > x₂ > isNat₁ > ok₁
proper₁ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
proper₁ > U11₃ > isNatKind₁ > ok₁
proper₁ > U11₃ > mark₁
proper₁ > U12₃ > U13₃ > isNatKind₁ > ok₁
proper₁ > U12₃ > U13₃ > mark₁
proper₁ > U14₃ > isNat₁ > ok₁
proper₁ > U14₃ > mark₁
proper₁ > U15₂ > isNat₁ > ok₁
proper₁ > U15₂ > U16₁ > mark₁
proper₁ > U15₂ > U16₁ > ok₁
proper₁ > U21₂ > isNatKind₁ > ok₁
proper₁ > U21₂ > mark₁
proper₁ > U31₃ > isNatKind₁ > ok₁
proper₁ > U31₃ > mark₁
proper₁ > U34₃ > U35₂ > isNat₁ > ok₁
proper₁ > U34₃ > U35₂ > mark₁
proper₁ > U41₂ > isNatKind₁ > ok₁
proper₁ > U41₂ > mark₁
proper₁ > U42₁ > tt > U16₁ > mark₁
proper₁ > U42₁ > tt > U16₁ > ok₁
proper₁ > U42₁ > tt > plus₂ > isNatKind₁ > ok₁
proper₁ > U42₁ > tt > plus₂ > isNat₁ > ok₁
proper₁ > U42₁ > tt > plus₂ > mark₁
proper₁ > U42₁ > tt > x₂ > isNatKind₁ > ok₁
proper₁ > U42₁ > tt > x₂ > isNat₁ > ok₁
proper₁ > U42₁ > tt > x₂ > mark₁
proper₁ > U42₁ > tt > U13₃ > isNatKind₁ > ok₁
proper₁ > U42₁ > tt > U13₃ > mark₁
proper₁ > U42₁ > tt > U22₂ > isNat₁ > ok₁
proper₁ > U42₁ > tt > U22₂ > mark₁
proper₁ > U42₁ > tt > U32₃ > isNatKind₁ > ok₁
proper₁ > U42₁ > tt > U32₃ > mark₁
proper₁ > U42₁ > tt > U33₃ > isNatKind₁ > ok₁
proper₁ > U42₁ > tt > U33₃ > mark₁
proper₁ > U42₁ > tt > U35₂ > isNat₁ > ok₁
proper₁ > U42₁ > tt > U35₂ > mark₁
proper₁ > U61₂ > isNatKind₁ > ok₁
proper₁ > U61₂ > mark₁
proper₁ > U71₂ > isNatKind₁ > ok₁
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U71₂ > U72₂ > ok₁
proper₁ > U81₃ > isNatKind₁ > ok₁
proper₁ > U81₃ > U82₃ > isNat₁ > ok₁
proper₁ > U81₃ > U82₃ > mark₁
proper₁ > U83₃ > isNatKind₁ > ok₁
proper₁ > U83₃ > mark₁
proper₁ > U84₃ > plus₂ > isNatKind₁ > ok₁
proper₁ > U84₃ > plus₂ > isNat₁ > ok₁
proper₁ > U84₃ > plus₂ > mark₁
proper₁ > 0 > isNat₁ > ok₁
proper₁ > 0 > U91₂ > isNatKind₁ > ok₁
proper₁ > 0 > U91₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(266) Obligation:

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

PROPER(U23(X)) → PROPER(X)
PROPER(U51(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U51(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U23(x1)  =  x1
U51(x1)  =  U51(x1)
U62(x1)  =  x1
s(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  x1
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U61(x1, x2)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U51₁ > tt > 0 > ok
active₁ > U51₁ > mark₁
active₁ > U102₃ > mark₁
active₁ > U102₃ > ok
active₁ > U103₃ > mark₁
active₁ > U103₃ > isNatKind > tt > 0 > ok
active₁ > U104₃ > mark₁
active₁ > U104₃ > ok
active₁ > plus₂ > isNat > mark₁
active₁ > plus₂ > isNat > isNatKind > tt > 0 > ok
active₁ > plus₂ > U81₃ > mark₁
active₁ > plus₂ > U81₃ > isNatKind > tt > 0 > ok
active₁ > x₂ > U101₃ > mark₁
active₁ > x₂ > U101₃ > isNatKind > tt > 0 > ok
active₁ > x₂ > isNat > mark₁
active₁ > x₂ > isNat > isNatKind > tt > 0 > ok
active₁ > x₂ > U31₁ > mark₁
active₁ > x₂ > U31₁ > isNatKind > tt > 0 > ok
active₁ > U11₁ > mark₁
active₁ > U11₁ > isNatKind > tt > 0 > ok
active₁ > U13₁ > mark₁
active₁ > U13₁ > isNatKind > tt > 0 > ok
active₁ > U22₁ > mark₁
active₁ > U22₁ > ok
active₁ > U32₁ > mark₁
active₁ > U32₁ > isNatKind > tt > 0 > ok
active₁ > U41₁ > mark₁
active₁ > U41₁ > isNatKind > tt > 0 > ok
active₁ > U42₁ > tt > 0 > ok
active₁ > U42₁ > mark₁
active₁ > U71₂ > isNatKind > tt > 0 > ok
active₁ > U71₂ > U72₂ > mark₁
active₁ > U71₂ > U72₂ > ok
active₁ > U82₃ > mark₁
active₁ > U82₃ > ok
active₁ > U83₃ > isNatKind > tt > 0 > ok
active₁ > U83₃ > U84₃ > mark₁
active₁ > U83₃ > U84₃ > ok
proper₁ > U51₁ > tt > 0 > ok
proper₁ > U51₁ > mark₁
proper₁ > U102₃ > mark₁
proper₁ > U102₃ > ok
proper₁ > U103₃ > mark₁
proper₁ > U103₃ > isNatKind > tt > 0 > ok
proper₁ > U104₃ > mark₁
proper₁ > U104₃ > ok
proper₁ > plus₂ > isNat > mark₁
proper₁ > plus₂ > isNat > isNatKind > tt > 0 > ok
proper₁ > plus₂ > U81₃ > mark₁
proper₁ > plus₂ > U81₃ > isNatKind > tt > 0 > ok
proper₁ > x₂ > U101₃ > mark₁
proper₁ > x₂ > U101₃ > isNatKind > tt > 0 > ok
proper₁ > x₂ > isNat > mark₁
proper₁ > x₂ > isNat > isNatKind > tt > 0 > ok
proper₁ > x₂ > U31₁ > mark₁
proper₁ > x₂ > U31₁ > isNatKind > tt > 0 > ok
proper₁ > U11₁ > mark₁
proper₁ > U11₁ > isNatKind > tt > 0 > ok
proper₁ > U13₁ > mark₁
proper₁ > U13₁ > isNatKind > tt > 0 > ok
proper₁ > U22₁ > mark₁
proper₁ > U22₁ > ok
proper₁ > U32₁ > mark₁
proper₁ > U32₁ > isNatKind > tt > 0 > ok
proper₁ > U41₁ > mark₁
proper₁ > U41₁ > isNatKind > tt > 0 > ok
proper₁ > U42₁ > tt > 0 > ok
proper₁ > U42₁ > mark₁
proper₁ > U71₂ > isNatKind > tt > 0 > ok
proper₁ > U71₂ > U72₂ > mark₁
proper₁ > U71₂ > U72₂ > ok
proper₁ > U82₃ > mark₁
proper₁ > U82₃ > ok
proper₁ > U83₃ > isNatKind > tt > 0 > ok
proper₁ > U83₃ > U84₃ > mark₁
proper₁ > U83₃ > U84₃ > ok

The following usable rules [FROCOS05] were oriented:

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

(268) Obligation:

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

PROPER(U23(X)) → PROPER(X)
PROPER(U62(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(269) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U23(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
U23(x1)  =  U23(x1)
U62(x1)  =  x1
s(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U101₃ > U102₃ > U103₃ > mark₁
active₁ > U104₃ > x₂ > mark₁
active₁ > plus₂ > U11₃ > mark₁
active₁ > plus₂ > U71₂ > U72₂ > mark₁
active₁ > plus₂ > U81₃ > mark₁
active₁ > U12₃ > U13₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > U23₁ > mark₁
active₁ > U14₃ > isNat₁ > tt > U13₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > U16₁ > mark₁
active₁ > U14₃ > isNat₁ > tt > U72₂ > mark₁
active₁ > U14₃ > isNat₁ > tt > U83₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > 0 > mark₁
active₁ > U14₃ > isNat₁ > U11₃ > mark₁
active₁ > U14₃ > isNat₁ > U21₂ > U22₂ > U23₁ > mark₁
active₁ > U14₃ > isNat₁ > U31₃ > U32₃ > mark₁
active₁ > U14₃ > U15₂ > U16₁ > mark₁
active₁ > U33₃ > U34₃ > U35₂ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > U42₁ > tt > U23₁ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > U42₁ > tt > U13₃ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > U42₁ > tt > U16₁ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > U42₁ > tt > U72₂ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > U42₁ > tt > U83₃ > mark₁
active₁ > U61₂ > isNatKind₁ > U41₂ > U42₁ > tt > 0 > mark₁
active₁ > U82₃ > U83₃ > mark₁
active₁ > U84₃ > mark₁
active₁ > U91₂ > mark₁
active₁ > U92₁ > 0 > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(270) Obligation:

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

PROPER(U62(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(271) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U62(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U62(x1)  =  U62(x1)
s(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  U51(x1)
U61(x1, x2)  =  U61(x1, x2)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U62₁ > tt > U102₃ > U103₃ > mark₁
active₁ > U62₁ > tt > U102₃ > isNat₁ > U11₃ > mark₁
active₁ > U62₁ > tt > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > U62₁ > tt > plus₂ > U81₃ > mark₁
active₁ > U62₁ > tt > x₂ > isNat₁ > U11₃ > mark₁
active₁ > U62₁ > tt > x₂ > U31₃ > mark₁
active₁ > U62₁ > tt > U12₃ > mark₁
active₁ > U62₁ > tt > U13₃ > U14₃ > mark₁
active₁ > U62₁ > tt > U15₂ > isNat₁ > U11₃ > mark₁
active₁ > U62₁ > tt > U16₁ > mark₁
active₁ > U62₁ > tt > U32₃ > mark₁
active₁ > U62₁ > tt > U35₂ > mark₁
active₁ > U62₁ > tt > U72₂ > mark₁
active₁ > U62₁ > tt > U82₃ > mark₁
active₁ > U62₁ > tt > U83₃ > mark₁
active₁ > U22₂ > ok > U101₃ > mark₁
active₁ > U22₂ > ok > U102₃ > U103₃ > mark₁
active₁ > U22₂ > ok > U102₃ > isNat₁ > U11₃ > mark₁
active₁ > U22₂ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > U22₂ > ok > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U22₂ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
active₁ > U22₂ > ok > U104₃ > x₂ > U31₃ > mark₁
active₁ > U22₂ > ok > U12₃ > mark₁
active₁ > U22₂ > ok > U13₃ > U14₃ > mark₁
active₁ > U22₂ > ok > U15₂ > isNat₁ > U11₃ > mark₁
active₁ > U22₂ > ok > U16₁ > mark₁
active₁ > U22₂ > ok > U21₂ > mark₁
active₁ > U22₂ > ok > U32₃ > mark₁
active₁ > U22₂ > ok > U35₂ > mark₁
active₁ > U22₂ > ok > U51₁ > mark₁
active₁ > U22₂ > ok > U61₂ > mark₁
active₁ > U22₂ > ok > U72₂ > mark₁
active₁ > U22₂ > ok > U82₃ > mark₁
active₁ > U22₂ > ok > U83₃ > mark₁
active₁ > U22₂ > ok > U91₂ > mark₁
active₁ > U33₃ > mark₁
active₁ > U34₃ > ok > U101₃ > mark₁
active₁ > U34₃ > ok > U102₃ > U103₃ > mark₁
active₁ > U34₃ > ok > U102₃ > isNat₁ > U11₃ > mark₁
active₁ > U34₃ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > U34₃ > ok > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U34₃ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
active₁ > U34₃ > ok > U104₃ > x₂ > U31₃ > mark₁
active₁ > U34₃ > ok > U12₃ > mark₁
active₁ > U34₃ > ok > U13₃ > U14₃ > mark₁
active₁ > U34₃ > ok > U15₂ > isNat₁ > U11₃ > mark₁
active₁ > U34₃ > ok > U16₁ > mark₁
active₁ > U34₃ > ok > U21₂ > mark₁
active₁ > U34₃ > ok > U32₃ > mark₁
active₁ > U34₃ > ok > U35₂ > mark₁
active₁ > U34₃ > ok > U51₁ > mark₁
active₁ > U34₃ > ok > U61₂ > mark₁
active₁ > U34₃ > ok > U72₂ > mark₁
active₁ > U34₃ > ok > U82₃ > mark₁
active₁ > U34₃ > ok > U83₃ > mark₁
active₁ > U34₃ > ok > U91₂ > mark₁
active₁ > U41₂ > mark₁
active₁ > U71₂ > ok > U101₃ > mark₁
active₁ > U71₂ > ok > U102₃ > U103₃ > mark₁
active₁ > U71₂ > ok > U102₃ > isNat₁ > U11₃ > mark₁
active₁ > U71₂ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > U71₂ > ok > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U71₂ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
active₁ > U71₂ > ok > U104₃ > x₂ > U31₃ > mark₁
active₁ > U71₂ > ok > U12₃ > mark₁
active₁ > U71₂ > ok > U13₃ > U14₃ > mark₁
active₁ > U71₂ > ok > U15₂ > isNat₁ > U11₃ > mark₁
active₁ > U71₂ > ok > U16₁ > mark₁
active₁ > U71₂ > ok > U21₂ > mark₁
active₁ > U71₂ > ok > U32₃ > mark₁
active₁ > U71₂ > ok > U35₂ > mark₁
active₁ > U71₂ > ok > U51₁ > mark₁
active₁ > U71₂ > ok > U61₂ > mark₁
active₁ > U71₂ > ok > U72₂ > mark₁
active₁ > U71₂ > ok > U82₃ > mark₁
active₁ > U71₂ > ok > U83₃ > mark₁
active₁ > U71₂ > ok > U91₂ > mark₁
active₁ > U84₃ > ok > U101₃ > mark₁
active₁ > U84₃ > ok > U102₃ > U103₃ > mark₁
active₁ > U84₃ > ok > U102₃ > isNat₁ > U11₃ > mark₁
active₁ > U84₃ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > U84₃ > ok > U104₃ > plus₂ > U81₃ > mark₁
active₁ > U84₃ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
active₁ > U84₃ > ok > U104₃ > x₂ > U31₃ > mark₁
active₁ > U84₃ > ok > U12₃ > mark₁
active₁ > U84₃ > ok > U13₃ > U14₃ > mark₁
active₁ > U84₃ > ok > U15₂ > isNat₁ > U11₃ > mark₁
active₁ > U84₃ > ok > U16₁ > mark₁
active₁ > U84₃ > ok > U21₂ > mark₁
active₁ > U84₃ > ok > U32₃ > mark₁
active₁ > U84₃ > ok > U35₂ > mark₁
active₁ > U84₃ > ok > U51₁ > mark₁
active₁ > U84₃ > ok > U61₂ > mark₁
active₁ > U84₃ > ok > U72₂ > mark₁
active₁ > U84₃ > ok > U82₃ > mark₁
active₁ > U84₃ > ok > U83₃ > mark₁
active₁ > U84₃ > ok > U91₂ > mark₁
active₁ > 0 > tt > U102₃ > U103₃ > mark₁
active₁ > 0 > tt > U102₃ > isNat₁ > U11₃ > mark₁
active₁ > 0 > tt > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > 0 > tt > plus₂ > U81₃ > mark₁
active₁ > 0 > tt > x₂ > isNat₁ > U11₃ > mark₁
active₁ > 0 > tt > x₂ > U31₃ > mark₁
active₁ > 0 > tt > U12₃ > mark₁
active₁ > 0 > tt > U13₃ > U14₃ > mark₁
active₁ > 0 > tt > U15₂ > isNat₁ > U11₃ > mark₁
active₁ > 0 > tt > U16₁ > mark₁
active₁ > 0 > tt > U32₃ > mark₁
active₁ > 0 > tt > U35₂ > mark₁
active₁ > 0 > tt > U72₂ > mark₁
active₁ > 0 > tt > U82₃ > mark₁
active₁ > 0 > tt > U83₃ > mark₁
active₁ > 0 > U91₂ > mark₁
proper₁ > U62₁ > tt > U102₃ > U103₃ > mark₁
proper₁ > U62₁ > tt > U102₃ > isNat₁ > U11₃ > mark₁
proper₁ > U62₁ > tt > plus₂ > isNat₁ > U11₃ > mark₁
proper₁ > U62₁ > tt > plus₂ > U81₃ > mark₁
proper₁ > U62₁ > tt > x₂ > isNat₁ > U11₃ > mark₁
proper₁ > U62₁ > tt > x₂ > U31₃ > mark₁
proper₁ > U62₁ > tt > U12₃ > mark₁
proper₁ > U62₁ > tt > U13₃ > U14₃ > mark₁
proper₁ > U62₁ > tt > U15₂ > isNat₁ > U11₃ > mark₁
proper₁ > U62₁ > tt > U16₁ > mark₁
proper₁ > U62₁ > tt > U32₃ > mark₁
proper₁ > U62₁ > tt > U35₂ > mark₁
proper₁ > U62₁ > tt > U72₂ > mark₁
proper₁ > U62₁ > tt > U82₃ > mark₁
proper₁ > U62₁ > tt > U83₃ > mark₁
proper₁ > U22₂ > ok > U101₃ > mark₁
proper₁ > U22₂ > ok > U102₃ > U103₃ > mark₁
proper₁ > U22₂ > ok > U102₃ > isNat₁ > U11₃ > mark₁
proper₁ > U22₂ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
proper₁ > U22₂ > ok > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U22₂ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
proper₁ > U22₂ > ok > U104₃ > x₂ > U31₃ > mark₁
proper₁ > U22₂ > ok > U12₃ > mark₁
proper₁ > U22₂ > ok > U13₃ > U14₃ > mark₁
proper₁ > U22₂ > ok > U15₂ > isNat₁ > U11₃ > mark₁
proper₁ > U22₂ > ok > U16₁ > mark₁
proper₁ > U22₂ > ok > U21₂ > mark₁
proper₁ > U22₂ > ok > U32₃ > mark₁
proper₁ > U22₂ > ok > U35₂ > mark₁
proper₁ > U22₂ > ok > U51₁ > mark₁
proper₁ > U22₂ > ok > U61₂ > mark₁
proper₁ > U22₂ > ok > U72₂ > mark₁
proper₁ > U22₂ > ok > U82₃ > mark₁
proper₁ > U22₂ > ok > U83₃ > mark₁
proper₁ > U22₂ > ok > U91₂ > mark₁
proper₁ > U33₃ > mark₁
proper₁ > U34₃ > ok > U101₃ > mark₁
proper₁ > U34₃ > ok > U102₃ > U103₃ > mark₁
proper₁ > U34₃ > ok > U102₃ > isNat₁ > U11₃ > mark₁
proper₁ > U34₃ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
proper₁ > U34₃ > ok > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U34₃ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
proper₁ > U34₃ > ok > U104₃ > x₂ > U31₃ > mark₁
proper₁ > U34₃ > ok > U12₃ > mark₁
proper₁ > U34₃ > ok > U13₃ > U14₃ > mark₁
proper₁ > U34₃ > ok > U15₂ > isNat₁ > U11₃ > mark₁
proper₁ > U34₃ > ok > U16₁ > mark₁
proper₁ > U34₃ > ok > U21₂ > mark₁
proper₁ > U34₃ > ok > U32₃ > mark₁
proper₁ > U34₃ > ok > U35₂ > mark₁
proper₁ > U34₃ > ok > U51₁ > mark₁
proper₁ > U34₃ > ok > U61₂ > mark₁
proper₁ > U34₃ > ok > U72₂ > mark₁
proper₁ > U34₃ > ok > U82₃ > mark₁
proper₁ > U34₃ > ok > U83₃ > mark₁
proper₁ > U34₃ > ok > U91₂ > mark₁
proper₁ > U41₂ > mark₁
proper₁ > U71₂ > ok > U101₃ > mark₁
proper₁ > U71₂ > ok > U102₃ > U103₃ > mark₁
proper₁ > U71₂ > ok > U102₃ > isNat₁ > U11₃ > mark₁
proper₁ > U71₂ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
proper₁ > U71₂ > ok > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U71₂ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
proper₁ > U71₂ > ok > U104₃ > x₂ > U31₃ > mark₁
proper₁ > U71₂ > ok > U12₃ > mark₁
proper₁ > U71₂ > ok > U13₃ > U14₃ > mark₁
proper₁ > U71₂ > ok > U15₂ > isNat₁ > U11₃ > mark₁
proper₁ > U71₂ > ok > U16₁ > mark₁
proper₁ > U71₂ > ok > U21₂ > mark₁
proper₁ > U71₂ > ok > U32₃ > mark₁
proper₁ > U71₂ > ok > U35₂ > mark₁
proper₁ > U71₂ > ok > U51₁ > mark₁
proper₁ > U71₂ > ok > U61₂ > mark₁
proper₁ > U71₂ > ok > U72₂ > mark₁
proper₁ > U71₂ > ok > U82₃ > mark₁
proper₁ > U71₂ > ok > U83₃ > mark₁
proper₁ > U71₂ > ok > U91₂ > mark₁
proper₁ > U84₃ > ok > U101₃ > mark₁
proper₁ > U84₃ > ok > U102₃ > U103₃ > mark₁
proper₁ > U84₃ > ok > U102₃ > isNat₁ > U11₃ > mark₁
proper₁ > U84₃ > ok > U104₃ > plus₂ > isNat₁ > U11₃ > mark₁
proper₁ > U84₃ > ok > U104₃ > plus₂ > U81₃ > mark₁
proper₁ > U84₃ > ok > U104₃ > x₂ > isNat₁ > U11₃ > mark₁
proper₁ > U84₃ > ok > U104₃ > x₂ > U31₃ > mark₁
proper₁ > U84₃ > ok > U12₃ > mark₁
proper₁ > U84₃ > ok > U13₃ > U14₃ > mark₁
proper₁ > U84₃ > ok > U15₂ > isNat₁ > U11₃ > mark₁
proper₁ > U84₃ > ok > U16₁ > mark₁
proper₁ > U84₃ > ok > U21₂ > mark₁
proper₁ > U84₃ > ok > U32₃ > mark₁
proper₁ > U84₃ > ok > U35₂ > mark₁
proper₁ > U84₃ > ok > U51₁ > mark₁
proper₁ > U84₃ > ok > U61₂ > mark₁
proper₁ > U84₃ > ok > U72₂ > mark₁
proper₁ > U84₃ > ok > U82₃ > mark₁
proper₁ > U84₃ > ok > U83₃ > mark₁
proper₁ > U84₃ > ok > U91₂ > mark₁
proper₁ > 0 > tt > U102₃ > U103₃ > mark₁
proper₁ > 0 > tt > U102₃ > isNat₁ > U11₃ > mark₁
proper₁ > 0 > tt > plus₂ > isNat₁ > U11₃ > mark₁
proper₁ > 0 > tt > plus₂ > U81₃ > mark₁
proper₁ > 0 > tt > x₂ > isNat₁ > U11₃ > mark₁
proper₁ > 0 > tt > x₂ > U31₃ > mark₁
proper₁ > 0 > tt > U12₃ > mark₁
proper₁ > 0 > tt > U13₃ > U14₃ > mark₁
proper₁ > 0 > tt > U15₂ > isNat₁ > U11₃ > mark₁
proper₁ > 0 > tt > U16₁ > mark₁
proper₁ > 0 > tt > U32₃ > mark₁
proper₁ > 0 > tt > U35₂ > mark₁
proper₁ > 0 > tt > U72₂ > mark₁
proper₁ > 0 > tt > U82₃ > mark₁
proper₁ > 0 > tt > U83₃ > mark₁
proper₁ > 0 > U91₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(272) Obligation:

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

PROPER(s(X)) → PROPER(X)

The TRS R consists of the following rules:

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

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

(273) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
s(x1)  =  s(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > isNatKind > U41₁ > U42₁ > tt > s₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > isNatKind > U41₁ > U42₁ > tt > U15₁ > mark₁
top > active₁ > isNatKind > U41₁ > U42₁ > tt > U32₁ > mark₁
top > active₁ > isNatKind > U61₁ > mark₁
top > active₁ > isNat > tt > s₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > isNat > tt > U15₁ > mark₁
top > active₁ > isNat > tt > U32₁ > mark₁
top > active₁ > isNat > U31₁ > U32₁ > mark₁
top > active₁ > plus₂ > U41₁ > U42₁ > tt > s₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > plus₂ > U41₁ > U42₁ > tt > U15₁ > mark₁
top > active₁ > plus₂ > U41₁ > U42₁ > tt > U32₁ > mark₁
top > active₁ > plus₂ > U71₂ > mark₁
top > active₁ > plus₂ > U81₃ > mark₁
top > active₁ > U13₁ > mark₁
top > active₁ > U23₁ > tt > s₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > U23₁ > tt > U15₁ > mark₁
top > active₁ > U23₁ > tt > U32₁ > mark₁
top > active₁ > U33₁ > mark₁
top > active₁ > U34₁ > mark₁
top > active₁ > U72₂ > mark₁
top > active₁ > U82₃ > mark₁
top > active₁ > U83₃ > U84₃ > s₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > 0 > tt > s₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > 0 > tt > U15₁ > mark₁
top > active₁ > 0 > tt > U32₁ > mark₁
top > active₁ > 0 > U71₂ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(274) Obligation:

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

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

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

(275) PisEmptyProof (EQUIVALENT transformation)

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

(276) TRUE

(277) Obligation:

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

ACTIVE(U102(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U101(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U103(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U104(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U12(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U13(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U14(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U15(X1, X2)) → ACTIVE(X1)
ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
ACTIVE(U23(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U32(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U33(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U34(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U35(X1, X2)) → ACTIVE(X1)
ACTIVE(U36(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U82(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U83(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U84(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(U92(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(278) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U104(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U102(x1, x2, x3)  =  x1
U101(x1, x2, x3)  =  x1
U103(x1, x2, x3)  =  x1
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  x1
U33(x1, x2, x3)  =  x1
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  x1
U72(x1, x2)  =  x1
U81(x1, x2, x3)  =  x1
U82(x1, x2, x3)  =  x1
U83(x1, x2, x3)  =  x1
U84(x1, x2, x3)  =  x1
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  mark
isNatKind(x1)  =  x1
isNat(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

tt > 0 > ok > active₁ > mark > U104₃
tt > 0 > ok > active₁ > mark > plus₂
tt > 0 > ok > active₁ > mark > x₂
top > active₁ > mark > U104₃
top > active₁ > mark > plus₂
top > active₁ > mark > x₂
top > proper₁ > U104₃
top > proper₁ > plus₂
top > proper₁ > x₂

The following usable rules [FROCOS05] were oriented:

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

(279) Obligation:

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

ACTIVE(U102(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U101(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U103(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U12(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U13(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U14(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U15(X1, X2)) → ACTIVE(X1)
ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
ACTIVE(U23(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U32(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U33(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U34(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U35(X1, X2)) → ACTIVE(X1)
ACTIVE(U36(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U82(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U83(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U84(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)
ACTIVE(U92(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(280) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U102(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U101(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U103(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U12(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U13(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U14(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U15(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
ACTIVE(U23(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U32(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U33(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U34(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U35(X1, X2)) → ACTIVE(X1)
ACTIVE(U36(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2)) → ACTIVE(X1)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U82(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U83(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U84(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U91(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U102(x1, x2, x3)  =  U102(x1, x2, x3)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
U51(x1)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  x1
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  x1
isNatKind(x1)  =  x1
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U11₃ > ok > U23₁ > U101₃
active₁ > U11₃ > ok > U33₃ > U101₃
active₁ > U11₃ > ok > U83₃ > U101₃
active₁ > U11₃ > ok > U84₃ > U101₃
active₁ > U12₃ > U13₃ > ok > U23₁ > U101₃
active₁ > U12₃ > U13₃ > ok > U33₃ > U101₃
active₁ > U12₃ > U13₃ > ok > U83₃ > U101₃
active₁ > U12₃ > U13₃ > ok > U84₃ > U101₃
active₁ > U14₃ > U15₂ > ok > U23₁ > U101₃
active₁ > U14₃ > U15₂ > ok > U33₃ > U101₃
active₁ > U14₃ > U15₂ > ok > U83₃ > U101₃
active₁ > U14₃ > U15₂ > ok > U84₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > U102₃ > U103₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > U102₃ > ok > U23₁ > U101₃
active₁ > U14₃ > isNat₁ > tt > U102₃ > ok > U33₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > U102₃ > ok > U83₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > U102₃ > ok > U84₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > U34₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > x₂ > U101₃
active₁ > U14₃ > isNat₁ > tt > 0 > ok > U23₁ > U101₃
active₁ > U14₃ > isNat₁ > tt > 0 > ok > U33₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > 0 > ok > U83₃ > U101₃
active₁ > U14₃ > isNat₁ > tt > 0 > ok > U84₃ > U101₃
active₁ > U21₂ > U22₂ > ok > U23₁ > U101₃
active₁ > U21₂ > U22₂ > ok > U33₃ > U101₃
active₁ > U21₂ > U22₂ > ok > U83₃ > U101₃
active₁ > U21₂ > U22₂ > ok > U84₃ > U101₃
active₁ > U31₃ > ok > U23₁ > U101₃
active₁ > U31₃ > ok > U33₃ > U101₃
active₁ > U31₃ > ok > U83₃ > U101₃
active₁ > U31₃ > ok > U84₃ > U101₃
active₁ > U32₃ > ok > U23₁ > U101₃
active₁ > U32₃ > ok > U33₃ > U101₃
active₁ > U32₃ > ok > U83₃ > U101₃
active₁ > U32₃ > ok > U84₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > U102₃ > U103₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > U102₃ > ok > U23₁ > U101₃
active₁ > U35₂ > isNat₁ > tt > U102₃ > ok > U33₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > U102₃ > ok > U83₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > U102₃ > ok > U84₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > U34₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > x₂ > U101₃
active₁ > U35₂ > isNat₁ > tt > 0 > ok > U23₁ > U101₃
active₁ > U35₂ > isNat₁ > tt > 0 > ok > U33₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > 0 > ok > U83₃ > U101₃
active₁ > U35₂ > isNat₁ > tt > 0 > ok > U84₃ > U101₃
active₁ > U36₁ > tt > U102₃ > U103₃ > U101₃
active₁ > U36₁ > tt > U102₃ > ok > U23₁ > U101₃
active₁ > U36₁ > tt > U102₃ > ok > U33₃ > U101₃
active₁ > U36₁ > tt > U102₃ > ok > U83₃ > U101₃
active₁ > U36₁ > tt > U102₃ > ok > U84₃ > U101₃
active₁ > U36₁ > tt > U34₃ > U101₃
active₁ > U36₁ > tt > x₂ > U101₃
active₁ > U36₁ > tt > 0 > ok > U23₁ > U101₃
active₁ > U36₁ > tt > 0 > ok > U33₃ > U101₃
active₁ > U36₁ > tt > 0 > ok > U83₃ > U101₃
active₁ > U36₁ > tt > 0 > ok > U84₃ > U101₃
active₁ > U41₂ > ok > U23₁ > U101₃
active₁ > U41₂ > ok > U33₃ > U101₃
active₁ > U41₂ > ok > U83₃ > U101₃
active₁ > U41₂ > ok > U84₃ > U101₃
active₁ > U61₂ > U101₃
active₁ > U71₂ > ok > U23₁ > U101₃
active₁ > U71₂ > ok > U33₃ > U101₃
active₁ > U71₂ > ok > U83₃ > U101₃
active₁ > U71₂ > ok > U84₃ > U101₃
active₁ > U72₂ > ok > U23₁ > U101₃
active₁ > U72₂ > ok > U33₃ > U101₃
active₁ > U72₂ > ok > U83₃ > U101₃
active₁ > U72₂ > ok > U84₃ > U101₃
active₁ > U81₃ > ok > U23₁ > U101₃
active₁ > U81₃ > ok > U33₃ > U101₃
active₁ > U81₃ > ok > U83₃ > U101₃
active₁ > U81₃ > ok > U84₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > U102₃ > U103₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > U102₃ > ok > U23₁ > U101₃
active₁ > U82₃ > isNat₁ > tt > U102₃ > ok > U33₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > U102₃ > ok > U83₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > U102₃ > ok > U84₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > U34₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > x₂ > U101₃
active₁ > U82₃ > isNat₁ > tt > 0 > ok > U23₁ > U101₃
active₁ > U82₃ > isNat₁ > tt > 0 > ok > U33₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > 0 > ok > U83₃ > U101₃
active₁ > U82₃ > isNat₁ > tt > 0 > ok > U84₃ > U101₃
active₁ > U91₂ > ok > U23₁ > U101₃
active₁ > U91₂ > ok > U33₃ > U101₃
active₁ > U91₂ > ok > U83₃ > U101₃
active₁ > U91₂ > ok > U84₃ > U101₃
active₁ > U104₃ > x₂ > U101₃
active₁ > U104₃ > ok > U23₁ > U101₃
active₁ > U104₃ > ok > U33₃ > U101₃
active₁ > U104₃ > ok > U83₃ > U101₃
active₁ > U104₃ > ok > U84₃ > U101₃
active₁ > plus₂ > ok > U23₁ > U101₃
active₁ > plus₂ > ok > U33₃ > U101₃
active₁ > plus₂ > ok > U83₃ > U101₃
active₁ > plus₂ > ok > U84₃ > U101₃
top > U101₃

The following usable rules [FROCOS05] were oriented:

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

(281) Obligation:

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

ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X)) → ACTIVE(X)
ACTIVE(U62(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U92(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(282) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U16(x1)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U62(x1)  =  x1
s(x1)  =  s(x1)
U92(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U61(x1, x2)  =  U61(x1, x2)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U91(x1, x2)  =  U91(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

ACTIVE₁ > mark₁
active₁ > U102₃ > mark₁
active₁ > isNatKind₁ > tt > mark₁
active₁ > isNatKind₁ > U41₂ > mark₁
active₁ > U103₃ > U104₃ > plus₂ > mark₁
active₁ > isNat₁ > tt > mark₁
active₁ > isNat₁ > U11₃ > mark₁
active₁ > isNat₁ > U21₂ > mark₁
active₁ > isNat₁ > U31₃ > U32₃ > mark₁
active₁ > x₂ > U101₃ > mark₁
active₁ > U12₃ > U13₃ > mark₁
active₁ > U14₃ > mark₁
active₁ > U15₂ > mark₁
active₁ > U22₂ > mark₁
active₁ > U23₁ > tt > mark₁
active₁ > U33₃ > mark₁
active₁ > U34₃ > U35₂ > mark₁
active₁ > U61₂ > mark₁
active₁ > U72₂ > mark₁
active₁ > U81₃ > U82₃ > mark₁
active₁ > U83₃ > mark₁
active₁ > U84₃ > s₁ > U101₃ > mark₁
active₁ > U84₃ > s₁ > U21₂ > mark₁
active₁ > U84₃ > plus₂ > mark₁
active₁ > 0 > tt > mark₁
active₁ > 0 > U71₂ > mark₁
active₁ > 0 > U91₂ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(283) Obligation:

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

ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X)) → ACTIVE(X)
ACTIVE(U62(X)) → ACTIVE(X)
ACTIVE(U92(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(284) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U92(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U16(x1)  =  x1
U42(x1)  =  x1
U51(x1)  =  x1
U62(x1)  =  x1
U92(x1)  =  U92(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  x1
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  x1
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  x1
U104(x1, x2, x3)  =  U104(x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x2, x3)
U12(x1, x2, x3)  =  U12(x2, x3)
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22
U23(x1)  =  U23
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  U41(x1, x2)
U61(x1, x2)  =  U61(x2)
U71(x1, x2)  =  x2
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x2, x3)
U84(x1, x2, x3)  =  U84(x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  x2
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U92₁ > ACTIVE₁ > U11₂
active₁ > U103₃ > U11₂
active₁ > U104₂ > U11₂
active₁ > plus₂ > U11₂
active₁ > x₂ > U101₃ > U102₃ > U11₂
active₁ > x₂ > U31₃ > U11₂
active₁ > U15₁ > U11₂
active₁ > U23 > U11₂
active₁ > U32₃ > U11₂
active₁ > U33₃ > U11₂
active₁ > U34₃ > U35₂ > U36₁ > U11₂
active₁ > U41₂ > U11₂
active₁ > U61₁ > U11₂
active₁ > U72₂ > U11₂
active₁ > U81₂ > U11₂
active₁ > U82₃ > U11₂
active₁ > U83₂ > U84₂ > U11₂
active₁ > s₁ > U101₃ > U102₃ > U11₂
active₁ > 0 > tt > U12₂ > U11₂
active₁ > 0 > tt > U14₁ > U11₂
active₁ > 0 > tt > U22 > U11₂
active₁ > 0 > tt > U36₁ > U11₂
active₁ > 0 > tt > U84₂ > U11₂
proper₁ > U92₁ > ACTIVE₁ > U11₂
proper₁ > U103₃ > U11₂
proper₁ > U104₂ > U11₂
proper₁ > plus₂ > U11₂
proper₁ > x₂ > U101₃ > U102₃ > U11₂
proper₁ > x₂ > U31₃ > U11₂
proper₁ > U15₁ > U11₂
proper₁ > U23 > U11₂
proper₁ > U32₃ > U11₂
proper₁ > U33₃ > U11₂
proper₁ > U34₃ > U35₂ > U36₁ > U11₂
proper₁ > U41₂ > U11₂
proper₁ > U61₁ > U11₂
proper₁ > U72₂ > U11₂
proper₁ > U81₂ > U11₂
proper₁ > U82₃ > U11₂
proper₁ > U83₂ > U84₂ > U11₂
proper₁ > s₁ > U101₃ > U102₃ > U11₂
proper₁ > 0 > tt > U12₂ > U11₂
proper₁ > 0 > tt > U14₁ > U11₂
proper₁ > 0 > tt > U22 > U11₂
proper₁ > 0 > tt > U36₁ > U11₂
proper₁ > 0 > tt > U84₂ > U11₂
top > U11₂

The following usable rules [FROCOS05] were oriented:

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

(285) Obligation:

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

ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X)) → ACTIVE(X)
ACTIVE(U62(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(286) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U16(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U16(x1)  =  U16(x1)
U42(x1)  =  x1
U51(x1)  =  x1
U62(x1)  =  x1
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat(x1)
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
U32(x1, x2, x3)  =  U32(x1, x2, x3)
U33(x1, x2, x3)  =  U33(x1, x2, x3)
U34(x1, x2, x3)  =  U34(x1, x2, x3)
U35(x1, x2)  =  U35(x1, x2)
U36(x1)  =  x1
U41(x1, x2)  =  U41(x1)
U61(x1, x2)  =  U61(x1)
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

active₁ > U16₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U16₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U16₁ > tt > U13₃ > mark₁
active₁ > U16₁ > tt > U13₃ > ok
active₁ > U16₁ > tt > U15₂ > mark₁
active₁ > U16₁ > tt > U15₂ > ok
active₁ > U16₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U16₁ > tt > U32₃ > U33₃ > ok
active₁ > U16₁ > tt > U72₂ > mark₁
active₁ > U16₁ > tt > U72₂ > ok
active₁ > U16₁ > tt > U83₃ > mark₁
active₁ > U16₁ > tt > U83₃ > ok
active₁ > U16₁ > tt > U92₁ > 0 > mark₁
active₁ > U16₁ > tt > U92₁ > 0 > ok
active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > mark₁
active₁ > x₂ > U101₃ > U102₃ > U103₃ > U104₃ > ok
active₁ > U12₃ > isNatKind > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U12₃ > isNatKind > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U12₃ > isNatKind > tt > U13₃ > mark₁
active₁ > U12₃ > isNatKind > tt > U13₃ > ok
active₁ > U12₃ > isNatKind > tt > U15₂ > mark₁
active₁ > U12₃ > isNatKind > tt > U15₂ > ok
active₁ > U12₃ > isNatKind > tt > U32₃ > U33₃ > mark₁
active₁ > U12₃ > isNatKind > tt > U32₃ > U33₃ > ok
active₁ > U12₃ > isNatKind > tt > U72₂ > mark₁
active₁ > U12₃ > isNatKind > tt > U72₂ > ok
active₁ > U12₃ > isNatKind > tt > U83₃ > mark₁
active₁ > U12₃ > isNatKind > tt > U83₃ > ok
active₁ > U12₃ > isNatKind > tt > U92₁ > 0 > mark₁
active₁ > U12₃ > isNatKind > tt > U92₁ > 0 > ok
active₁ > U12₃ > isNatKind > U41₁ > mark₁
active₁ > U12₃ > isNatKind > U41₁ > ok
active₁ > U12₃ > isNatKind > U61₁ > mark₁
active₁ > U12₃ > isNatKind > U61₁ > ok
active₁ > U14₃ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U14₃ > isNat₁ > tt > U13₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > U13₃ > ok
active₁ > U14₃ > isNat₁ > tt > U15₂ > mark₁
active₁ > U14₃ > isNat₁ > tt > U15₂ > ok
active₁ > U14₃ > isNat₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > U32₃ > U33₃ > ok
active₁ > U14₃ > isNat₁ > tt > U72₂ > mark₁
active₁ > U14₃ > isNat₁ > tt > U72₂ > ok
active₁ > U14₃ > isNat₁ > tt > U83₃ > mark₁
active₁ > U14₃ > isNat₁ > tt > U83₃ > ok
active₁ > U14₃ > isNat₁ > tt > U92₁ > 0 > mark₁
active₁ > U14₃ > isNat₁ > tt > U92₁ > 0 > ok
active₁ > U14₃ > isNat₁ > U11₃ > mark₁
active₁ > U14₃ > isNat₁ > U11₃ > ok
active₁ > U14₃ > isNat₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U14₃ > isNat₁ > U31₃ > U32₃ > U33₃ > ok
active₁ > U21₂ > isNatKind > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U21₂ > isNatKind > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U21₂ > isNatKind > tt > U13₃ > mark₁
active₁ > U21₂ > isNatKind > tt > U13₃ > ok
active₁ > U21₂ > isNatKind > tt > U15₂ > mark₁
active₁ > U21₂ > isNatKind > tt > U15₂ > ok
active₁ > U21₂ > isNatKind > tt > U32₃ > U33₃ > mark₁
active₁ > U21₂ > isNatKind > tt > U32₃ > U33₃ > ok
active₁ > U21₂ > isNatKind > tt > U72₂ > mark₁
active₁ > U21₂ > isNatKind > tt > U72₂ > ok
active₁ > U21₂ > isNatKind > tt > U83₃ > mark₁
active₁ > U21₂ > isNatKind > tt > U83₃ > ok
active₁ > U21₂ > isNatKind > tt > U92₁ > 0 > mark₁
active₁ > U21₂ > isNatKind > tt > U92₁ > 0 > ok
active₁ > U21₂ > isNatKind > U41₁ > mark₁
active₁ > U21₂ > isNatKind > U41₁ > ok
active₁ > U21₂ > isNatKind > U61₁ > mark₁
active₁ > U21₂ > isNatKind > U61₁ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U13₃ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U13₃ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U15₂ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U15₂ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U32₃ > U33₃ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U72₂ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U72₂ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U83₃ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U83₃ > ok
active₁ > U21₂ > U22₂ > isNat₁ > tt > U92₁ > 0 > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > tt > U92₁ > 0 > ok
active₁ > U21₂ > U22₂ > isNat₁ > U11₃ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > U11₃ > ok
active₁ > U21₂ > U22₂ > isNat₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U21₂ > U22₂ > isNat₁ > U31₃ > U32₃ > U33₃ > ok
active₁ > U23₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U23₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U23₁ > tt > U13₃ > mark₁
active₁ > U23₁ > tt > U13₃ > ok
active₁ > U23₁ > tt > U15₂ > mark₁
active₁ > U23₁ > tt > U15₂ > ok
active₁ > U23₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U23₁ > tt > U32₃ > U33₃ > ok
active₁ > U23₁ > tt > U72₂ > mark₁
active₁ > U23₁ > tt > U72₂ > ok
active₁ > U23₁ > tt > U83₃ > mark₁
active₁ > U23₁ > tt > U83₃ > ok
active₁ > U23₁ > tt > U92₁ > 0 > mark₁
active₁ > U23₁ > tt > U92₁ > 0 > ok
active₁ > U34₃ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U34₃ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U34₃ > isNat₁ > tt > U13₃ > mark₁
active₁ > U34₃ > isNat₁ > tt > U13₃ > ok
active₁ > U34₃ > isNat₁ > tt > U15₂ > mark₁
active₁ > U34₃ > isNat₁ > tt > U15₂ > ok
active₁ > U34₃ > isNat₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U34₃ > isNat₁ > tt > U32₃ > U33₃ > ok
active₁ > U34₃ > isNat₁ > tt > U72₂ > mark₁
active₁ > U34₃ > isNat₁ > tt > U72₂ > ok
active₁ > U34₃ > isNat₁ > tt > U83₃ > mark₁
active₁ > U34₃ > isNat₁ > tt > U83₃ > ok
active₁ > U34₃ > isNat₁ > tt > U92₁ > 0 > mark₁
active₁ > U34₃ > isNat₁ > tt > U92₁ > 0 > ok
active₁ > U34₃ > isNat₁ > U11₃ > mark₁
active₁ > U34₃ > isNat₁ > U11₃ > ok
active₁ > U34₃ > isNat₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U34₃ > isNat₁ > U31₃ > U32₃ > U33₃ > ok
active₁ > U35₂ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U35₂ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U35₂ > isNat₁ > tt > U13₃ > mark₁
active₁ > U35₂ > isNat₁ > tt > U13₃ > ok
active₁ > U35₂ > isNat₁ > tt > U15₂ > mark₁
active₁ > U35₂ > isNat₁ > tt > U15₂ > ok
active₁ > U35₂ > isNat₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U35₂ > isNat₁ > tt > U32₃ > U33₃ > ok
active₁ > U35₂ > isNat₁ > tt > U72₂ > mark₁
active₁ > U35₂ > isNat₁ > tt > U72₂ > ok
active₁ > U35₂ > isNat₁ > tt > U83₃ > mark₁
active₁ > U35₂ > isNat₁ > tt > U83₃ > ok
active₁ > U35₂ > isNat₁ > tt > U92₁ > 0 > mark₁
active₁ > U35₂ > isNat₁ > tt > U92₁ > 0 > ok
active₁ > U35₂ > isNat₁ > U11₃ > mark₁
active₁ > U35₂ > isNat₁ > U11₃ > ok
active₁ > U35₂ > isNat₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U35₂ > isNat₁ > U31₃ > U32₃ > U33₃ > ok
active₁ > U82₃ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U82₃ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U82₃ > isNat₁ > tt > U13₃ > mark₁
active₁ > U82₃ > isNat₁ > tt > U13₃ > ok
active₁ > U82₃ > isNat₁ > tt > U15₂ > mark₁
active₁ > U82₃ > isNat₁ > tt > U15₂ > ok
active₁ > U82₃ > isNat₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U82₃ > isNat₁ > tt > U32₃ > U33₃ > ok
active₁ > U82₃ > isNat₁ > tt > U72₂ > mark₁
active₁ > U82₃ > isNat₁ > tt > U72₂ > ok
active₁ > U82₃ > isNat₁ > tt > U83₃ > mark₁
active₁ > U82₃ > isNat₁ > tt > U83₃ > ok
active₁ > U82₃ > isNat₁ > tt > U92₁ > 0 > mark₁
active₁ > U82₃ > isNat₁ > tt > U92₁ > 0 > ok
active₁ > U82₃ > isNat₁ > U11₃ > mark₁
active₁ > U82₃ > isNat₁ > U11₃ > ok
active₁ > U82₃ > isNat₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U82₃ > isNat₁ > U31₃ > U32₃ > U33₃ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U13₃ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U13₃ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U15₂ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U15₂ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U32₃ > U33₃ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U32₃ > U33₃ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U72₂ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U72₂ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U83₃ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U83₃ > ok
active₁ > U84₃ > plus₂ > isNatKind > tt > U92₁ > 0 > mark₁
active₁ > U84₃ > plus₂ > isNatKind > tt > U92₁ > 0 > ok
active₁ > U84₃ > plus₂ > isNatKind > U41₁ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > U41₁ > ok
active₁ > U84₃ > plus₂ > isNatKind > U61₁ > mark₁
active₁ > U84₃ > plus₂ > isNatKind > U61₁ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U102₃ > U103₃ > U104₃ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U13₃ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U13₃ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U15₂ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U15₂ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U32₃ > U33₃ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U32₃ > U33₃ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U72₂ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U72₂ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U83₃ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U83₃ > ok
active₁ > U84₃ > plus₂ > isNat₁ > tt > U92₁ > 0 > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > tt > U92₁ > 0 > ok
active₁ > U84₃ > plus₂ > isNat₁ > U11₃ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > U11₃ > ok
active₁ > U84₃ > plus₂ > isNat₁ > U31₃ > U32₃ > U33₃ > mark₁
active₁ > U84₃ > plus₂ > isNat₁ > U31₃ > U32₃ > U33₃ > ok
active₁ > U84₃ > plus₂ > U71₂ > U72₂ > mark₁
active₁ > U84₃ > plus₂ > U71₂ > U72₂ > ok
active₁ > U84₃ > plus₂ > U81₃ > mark₁
active₁ > U84₃ > plus₂ > U81₃ > ok

The following usable rules [FROCOS05] were oriented:

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

(287) Obligation:

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

ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X)) → ACTIVE(X)
ACTIVE(U62(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(288) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U62(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U42(x1)  =  U42(x1)
U51(x1)  =  x1
U62(x1)  =  U62(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  x1
U12(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2, x3)  =  U31(x1)
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  U33(x1)
U34(x1, x2, x3)  =  U34(x1)
U35(x1, x2)  =  x1
U36(x1)  =  x1
U41(x1, x2)  =  x1
U61(x1, x2)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  x1
U91(x1, x2)  =  x1
U92(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

top > active₁ > U62₁ > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U62₁ > tt > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > U62₁ > tt > ok > U42₁ > mark₁
top > active₁ > U62₁ > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U62₁ > tt > ok > U104₃ > x₂ > mark₁
top > active₁ > U62₁ > tt > ok > U13₁ > U14₁ > mark₁
top > active₁ > U62₁ > tt > ok > U32₁ > mark₁
top > active₁ > U62₁ > tt > ok > U34₁ > mark₁
top > active₁ > U62₁ > tt > ok > U72₂ > mark₁
top > active₁ > U62₁ > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > active₁ > U101₃ > U102₃ > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U42₁ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U104₃ > x₂ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U13₁ > U14₁ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U32₁ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U34₁ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U72₂ > mark₁
top > active₁ > U101₃ > U102₃ > ok > U82₃ > U83₃ > U84₃ > mark₁
top > active₁ > U21₁ > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U21₁ > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U42₁ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U32₁ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U34₁ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U72₂ > mark₁
top > active₁ > U21₁ > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > active₁ > U31₁ > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U31₁ > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U42₁ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U32₁ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U34₁ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U72₂ > mark₁
top > active₁ > U31₁ > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > active₁ > U33₁ > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U33₁ > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U42₁ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U32₁ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U34₁ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U72₂ > mark₁
top > active₁ > U33₁ > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U42₁ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U32₁ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U34₁ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U72₂ > mark₁
top > active₁ > 0 > isNat > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > active₁ > 0 > U71₂ > ok > U42₁ > mark₁
top > active₁ > 0 > U71₂ > ok > U104₃ > plus₂ > U81₃ > mark₁
top > active₁ > 0 > U71₂ > ok > U104₃ > x₂ > mark₁
top > active₁ > 0 > U71₂ > ok > U13₁ > U14₁ > mark₁
top > active₁ > 0 > U71₂ > ok > U32₁ > mark₁
top > active₁ > 0 > U71₂ > ok > U34₁ > mark₁
top > active₁ > 0 > U71₂ > ok > U72₂ > mark₁
top > active₁ > 0 > U71₂ > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U62₁ > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U62₁ > tt > U103₃ > U104₃ > x₂ > mark₁
top > proper₁ > U62₁ > tt > ok > U42₁ > mark₁
top > proper₁ > U62₁ > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U62₁ > tt > ok > U104₃ > x₂ > mark₁
top > proper₁ > U62₁ > tt > ok > U13₁ > U14₁ > mark₁
top > proper₁ > U62₁ > tt > ok > U32₁ > mark₁
top > proper₁ > U62₁ > tt > ok > U34₁ > mark₁
top > proper₁ > U62₁ > tt > ok > U72₂ > mark₁
top > proper₁ > U62₁ > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U101₃ > U102₃ > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U101₃ > U102₃ > U103₃ > U104₃ > x₂ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U42₁ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U104₃ > x₂ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U13₁ > U14₁ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U32₁ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U34₁ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U72₂ > mark₁
top > proper₁ > U101₃ > U102₃ > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U42₁ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U32₁ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U34₁ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U72₂ > mark₁
top > proper₁ > U21₁ > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U42₁ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U32₁ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U34₁ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U72₂ > mark₁
top > proper₁ > U31₁ > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U42₁ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U32₁ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U34₁ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U72₂ > mark₁
top > proper₁ > U33₁ > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > U103₃ > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > U103₃ > U104₃ > x₂ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U42₁ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U104₃ > x₂ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U13₁ > U14₁ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U32₁ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U34₁ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U72₂ > mark₁
top > proper₁ > 0 > isNat > isNatKind > tt > ok > U82₃ > U83₃ > U84₃ > mark₁
top > proper₁ > 0 > U71₂ > ok > U42₁ > mark₁
top > proper₁ > 0 > U71₂ > ok > U104₃ > plus₂ > U81₃ > mark₁
top > proper₁ > 0 > U71₂ > ok > U104₃ > x₂ > mark₁
top > proper₁ > 0 > U71₂ > ok > U13₁ > U14₁ > mark₁
top > proper₁ > 0 > U71₂ > ok > U32₁ > mark₁
top > proper₁ > 0 > U71₂ > ok > U34₁ > mark₁
top > proper₁ > 0 > U71₂ > ok > U72₂ > mark₁
top > proper₁ > 0 > U71₂ > ok > U82₃ > U83₃ > U84₃ > mark₁

The following usable rules [FROCOS05] were oriented:

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

(289) Obligation:

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

ACTIVE(U51(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

(290) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U51(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U51(x1)  =  U51(x1)
active(x1)  =  active(x1)
U101(x1, x2, x3)  =  U101(x1, x2, x3)
tt  =  tt
mark(x1)  =  mark(x1)
U102(x1, x2, x3)  =  U102(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103(x1, x2, x3)
isNat(x1)  =  isNat
U104(x1, x2, x3)  =  U104(x1, x2, x3)
plus(x1, x2)  =  plus(x1, x2)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2, x3)  =  U11(x1)
U12(x1, x2, x3)  =  U12(x1)
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2, x3)  =  x1
U32(x1, x2, x3)  =  U32(x1)
U33(x1, x2, x3)  =  x1
U34(x1, x2, x3)  =  x1
U35(x1, x2)  =  U35(x1)
U36(x1)  =  U36(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
U72(x1, x2)  =  U72(x1, x2)
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
s(x1)  =  s(x1)
U91(x1, x2)  =  U91(x1)
U92(x1)  =  U92(x1)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:

ACTIVE₁ > mark₁
active₁ > U51₁ > tt > U23₁ > mark₁
active₁ > U51₁ > tt > U36₁ > mark₁
active₁ > U51₁ > tt > U84₃ > plus₂ > mark₁
active₁ > U102₃ > isNat > tt > U23₁ > mark₁
active₁ > U102₃ > isNat > tt > U36₁ > mark₁
active₁ > U102₃ > isNat > tt > U84₃ > plus₂ > mark₁
active₁ > isNatKind > tt > U23₁ > mark₁
active₁ > isNatKind > tt > U36₁ > mark₁
active₁ > isNatKind > tt > U84₃ > plus₂ > mark₁
active₁ > isNatKind > U61₁ > mark₁
active₁ > U103₃ > mark₁
active₁ > U104₃ > x₂ > isNat > tt > U23₁ > mark₁
active₁ > U104₃ > x₂ > isNat > tt > U36₁ > mark₁
active₁ > U104₃ > x₂ > isNat > tt > U84₃ > plus₂ > mark₁
active₁ > U11₁ > mark₁
active₁ > U12₁ > mark₁
active₁ > U14₁ > U15₁ > mark₁
active₁ > U16₁ > tt > U23₁ > mark₁
active₁ > U16₁ > tt > U36₁ > mark₁
active₁ > U16₁ > tt > U84₃ > plus₂ > mark₁
active₁ > U32₁ > mark₁
active₁ > U35₁ > isNat > tt > U23₁ > mark₁
active₁ > U35₁ > isNat > tt > U36₁ > mark₁
active₁ > U35₁ > isNat > tt > U84₃ > plus₂ > mark₁
active₁ > U72₂ > mark₁
active₁ > U82₃ > isNat > tt > U23₁ > mark₁
active₁ > U82₃ > isNat > tt > U36₁ > mark₁
active₁ > U82₃ > isNat > tt > U84₃ > plus₂ > mark₁
active₁ > U82₃ > U83₃ > U84₃ > plus₂ > mark₁
active₁ > s₁ > U101₃ > mark₁
active₁ > s₁ > isNat > tt > U23₁ > mark₁
active₁ > s₁ > isNat > tt > U36₁ > mark₁
active₁ > s₁ > isNat > tt > U84₃ > plus₂ > mark₁
active₁ > s₁ > U81₃ > mark₁
active₁ > U91₁ > U92₁ > mark₁
active₁ > 0 > isNat > tt > U23₁ > mark₁
active₁ > 0 > isNat > tt > U36₁ > mark₁
active₁ > 0 > isNat > tt > U84₃ > plus₂ > mark₁
active₁ > 0 > U71₂ > mark₁
top > mark₁

The following usable rules [FROCOS05] were oriented:

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

(291) Obligation:

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

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

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

(292) PisEmptyProof (EQUIVALENT transformation)

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

(293) TRUE

(294) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following rules:

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

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