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

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

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

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
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)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X)) → U41(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2, X3)) → U62(active(X1), X2, X3)
active(U63(X1, X2, X3)) → U63(active(X1), X2, X3)
active(U64(X1, X2, X3)) → U64(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(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) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X)) → mark(U41(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2, X3) → mark(U62(X1, X2, X3))
U63(mark(X1), X2, X3) → mark(U63(X1, X2, X3))
U64(mark(X1), X2, X3) → mark(U64(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
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(isNat(X)) → isNat(proper(X))
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)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X)) → U41(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2, X3)) → U62(proper(X1), proper(X2), proper(X3))
proper(U63(X1, X2, X3)) → U63(proper(X1), proper(X2), proper(X3))
proper(U64(X1, X2, X3)) → U64(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
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))
isNat(ok(X)) → ok(isNat(X))
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(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X)) → ok(U41(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2), ok(X3)) → ok(U62(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3)) → ok(U63(X1, X2, X3))
U64(ok(X1), ok(X2), ok(X3)) → ok(U64(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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.
The approximation of the Dependency Graph [15,17,22] contains 25 SCCs with 83 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
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)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X)) → U41(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2, X3)) → U62(active(X1), X2, X3)
active(U63(X1, X2, X3)) → U63(active(X1), X2, X3)
active(U64(X1, X2, X3)) → U64(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(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) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X)) → mark(U41(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2, X3) → mark(U62(X1, X2, X3))
U63(mark(X1), X2, X3) → mark(U63(X1, X2, X3))
U64(mark(X1), X2, X3) → mark(U64(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
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(isNat(X)) → isNat(proper(X))
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)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X)) → U41(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2, X3)) → U62(proper(X1), proper(X2), proper(X3))
proper(U63(X1, X2, X3)) → U63(proper(X1), proper(X2), proper(X3))
proper(U64(X1, X2, X3)) → U64(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
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))
isNat(ok(X)) → ok(isNat(X))
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(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X)) → ok(U41(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2), ok(X3)) → ok(U62(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3)) → ok(U63(X1, X2, X3))
U64(ok(X1), ok(X2), ok(X3)) → ok(U64(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNAT(ok(X)) → ISNAT(X)

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ISNATKIND(ok(X)) → ISNATKIND(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U41¹(ok(X)) → U41¹(X)
U41¹(mark(X)) → U41¹(X)

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U41¹(ok(X)) → U41¹(X)
U41¹(mark(X)) → U41¹(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U32¹(ok(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

U32¹(ok(X)) → U32¹(X)
U32¹(mark(X)) → U32¹(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

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

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

PROPER(U63(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(isNat(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U16(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U52(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U12(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U15(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U13(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U63(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U14(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U12(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U23(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U14(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U64(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U64(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U12(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(isNatKind(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U32(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U63(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U21(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U13(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(plus(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U64(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U14(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U15(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U52(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U21(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U22(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(plus(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U41(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U13(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U22(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

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

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

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U64(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U23(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U62(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U16(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U32(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U13(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U63(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U12(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U41(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U14(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U15(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
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)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X)) → U41(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2, X3)) → U62(active(X1), X2, X3)
active(U63(X1, X2, X3)) → U63(active(X1), X2, X3)
active(U64(X1, X2, X3)) → U64(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(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) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X)) → mark(U41(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2, X3) → mark(U62(X1, X2, X3))
U63(mark(X1), X2, X3) → mark(U63(X1, X2, X3))
U64(mark(X1), X2, X3) → mark(U64(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
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(isNat(X)) → isNat(proper(X))
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)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X)) → U41(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2, X3)) → U62(proper(X1), proper(X2), proper(X3))
proper(U63(X1, X2, X3)) → U63(proper(X1), proper(X2), proper(X3))
proper(U64(X1, X2, X3)) → U64(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
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))
isNat(ok(X)) → ok(isNat(X))
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(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X)) → ok(U41(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2), ok(X3)) → ok(U62(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3)) → ok(U63(X1, X2, X3))
U64(ok(X1), ok(X2), ok(X3)) → ok(U64(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U12(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U13(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U14(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U15(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + 2·x₂
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = x₁ + 2·x₂
POL(U32(x₁)) = x₁
POL(U41(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U63(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U64(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(active(x₁)) = 2·x₁
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 2·x₁
POL(plus(x₁, x₂)) = 2·x₁ + 2·x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = 2·x₁
POL(tt) = 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(X)) → TOP(active(X))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(U23(tt))) → TOP(mark(tt))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U16(tt))) → TOP(mark(tt))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(plus(x0, 0))) → TOP(mark(U51(isNat(x0), x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatKind(x0), x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(isNatKind(0))) → TOP(mark(tt))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U41(tt))) → TOP(mark(tt))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

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

TOP(ok(U23(tt))) → TOP(mark(tt))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U16(tt))) → TOP(mark(tt))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(plus(x0, 0))) → TOP(mark(U51(isNat(x0), x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(ok(isNatKind(0))) → TOP(mark(tt))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(tt))) → TOP(mark(tt))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(mark(tt)) → TOP(ok(tt))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatKind(x0), x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(plus(x0, 0))) → TOP(mark(U51(isNat(x0), x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatKind(x0), x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(plus(x0, 0))) → TOP(mark(U51(isNat(x0), x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatKind(x0), x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatKind(x0), x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatKind(x0), x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNatKind(x0), x0, x1)))
TOP(ok(U21(tt, x0))) → TOP(mark(U22(isNatKind(x0), x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNatKind(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U63(tt, x0, x1))) → TOP(mark(U64(isNatKind(x1), x0, x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNatKind(x0), x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U22(tt, x0))) → TOP(mark(U23(isNat(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNatKind(s(x0)))) → TOP(mark(U41(isNatKind(x0))))
TOP(ok(isNatKind(plus(x0, x1)))) → TOP(mark(U31(isNatKind(x0), x1)))
TOP(ok(U52(tt, x0))) → TOP(mark(x0))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U11(tt, x0, x1))) → TOP(mark(U12(isNatKind(x0), x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U12(tt, x0, x1))) → TOP(mark(U13(isNatKind(x1), x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(mark(isNatKind(x0))) → TOP(isNatKind(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

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

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(plus(x0, s(x1)))) → TOP(mark(U61(isNat(x1), x1, x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U64(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

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

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNatKind(x0), x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U13(tt, x0, x1))) → TOP(mark(U14(isNatKind(x1), x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

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

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U15(tt, x0))) → TOP(mark(U16(isNat(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U62(tt, x0, x1))) → TOP(mark(U63(isNat(x1), x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

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

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

TOP(ok(U14(tt, x0, x1))) → TOP(mark(U15(isNat(x0), x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

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

TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U13(x0, x1, x2))) → TOP(U13(active(x0), x1, x2))
TOP(ok(U11(x0, x1, x2))) → TOP(U11(active(x0), x1, x2))
TOP(ok(U12(x0, x1, x2))) → TOP(U12(active(x0), x1, x2))
TOP(mark(U15(x0, x1))) → TOP(U15(proper(x0), proper(x1)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U23(x0))) → TOP(U23(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U11(x0, x1, x2))) → TOP(U11(proper(x0), proper(x1), proper(x2)))
TOP(mark(U21(x0, x1))) → TOP(U21(proper(x0), proper(x1)))
TOP(ok(U15(x0, x1))) → TOP(U15(active(x0), x1))
TOP(ok(U22(x0, x1))) → TOP(U22(active(x0), x1))
TOP(ok(U64(x0, x1, x2))) → TOP(U64(active(x0), x1, x2))
TOP(ok(U16(x0))) → TOP(U16(active(x0)))
TOP(ok(U21(x0, x1))) → TOP(U21(active(x0), x1))
TOP(mark(U22(x0, x1))) → TOP(U22(proper(x0), proper(x1)))
TOP(mark(U64(x0, x1, x2))) → TOP(U64(proper(x0), proper(x1), proper(x2)))
TOP(mark(U14(x0, x1, x2))) → TOP(U14(proper(x0), proper(x1), proper(x2)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U52(x0, x1))) → TOP(U52(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U16(x0))) → TOP(U16(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U41(x0))) → TOP(U41(proper(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(mark(U62(x0, x1, x2))) → TOP(U62(proper(x0), proper(x1), proper(x2)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U13(x0, x1, x2))) → TOP(U13(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(x0, x1))) → TOP(U52(active(x0), x1))
TOP(mark(U63(x0, x1, x2))) → TOP(U63(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U12(x0, x1, x2))) → TOP(U12(proper(x0), proper(x1), proper(x2)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(ok(U14(x0, x1, x2))) → TOP(U14(active(x0), x1, x2))
TOP(ok(U62(x0, x1, x2))) → TOP(U62(active(x0), x1, x2))
TOP(ok(U63(x0, x1, x2))) → TOP(U63(active(x0), x1, x2))
TOP(mark(U23(x0))) → TOP(U23(proper(x0)))
TOP(ok(U41(x0))) → TOP(U41(active(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

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

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