(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 25 SCCs with 83 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(ok(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
tt  =  tt
mark(x1)  =  x1
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  U13(x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x2)
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ISNAT1 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U112, U123, U512, plus2] > U132 > U143 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U112, U123, U512, plus2] > U311 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U112, U123, U512, plus2] > U612 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > U151 > U161 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U211, s1] > [U221, U231] > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U211, s1] > U612 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > U642 > [ok1, U521]
top > [ok1, U521]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(7) Obligation:

Q DP problem:
P is empty.
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.

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNATKIND(ok(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(ok(X)) → ISNATKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
tt  =  tt
mark(x1)  =  x1
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  U13(x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x2)
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ISNATKIND1 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U112, U123, U512, plus2] > U132 > U143 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U112, U123, U512, plus2] > U311 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U112, U123, U512, plus2] > U612 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > U151 > U161 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U211, s1] > [U221, U231] > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > [U211, s1] > U612 > [ok1, U521]
proper1 > [tt, 0] > [active1, U321, U411, U622, U633] > U642 > [ok1, U521]
top > [ok1, U521]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(12) Obligation:

Q DP problem:
P is empty.
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.

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(mark(X1), X2) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > active1 > [U123, U152, U512, U522, proper1] > U113 > [isNatKind, U613, U633, U643, plus2] > tt > U143 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > U113 > [isNatKind, U613, U633, U643, plus2] > tt > U623 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > U113 > [isNatKind, U613, U633, U643, plus2] > U311 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > U133 > U143 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > [U212, U222] > [isNatKind, U613, U633, U643, plus2] > tt > U143 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > [U212, U222] > [isNatKind, U613, U633, U643, plus2] > tt > U623 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > [U212, U222] > [isNatKind, U613, U633, U643, plus2] > U311 > [mark1, U411]
top > active1 > [U123, U152, U512, U522, proper1] > 0 > [mark1, U411]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(17) Obligation:

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

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

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x1)
mark(x1)  =  x1
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
PLUS1 > [ok1, tt, isNatKind1, U131, U141, U151, U221, U321, 0]
top > proper1 > [active1, U121, U521] > [U112, U312, U612, U622, U632, U642, plus2] > [ok1, tt, isNatKind1, U131, U141, U151, U221, U321, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(19) Obligation:

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

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))

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(X1, mark(X2)) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  PLUS(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
isNat(x1)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U113 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U133 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U152 > U161 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U222 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U411 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U522 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U613 > [PLUS1, mark1]
[active1, tt, U123, isNatKind, 0, top] > [U143, U212, U512, plus2, ok] > U623 > [U633, U643] > [PLUS1, mark1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(21) Obligation:

Q DP problem:
P is empty.
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.

(22) PisEmptyProof (EQUIVALENT transformation)

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

(23) TRUE

(24) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(ok(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U141 > [ok1, U111, U121, U131, U221, U321] > S1
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U151 > [ok1, U111, U121, U131, U221, U321] > S1
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > [U211, s1] > U613 > [ok1, U111, U121, U131, U221, U321] > S1


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(26) Obligation:

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

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))

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U512, plus2, top] > U121 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U161 > tt > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U522 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U613 > U623 > U633 > U643 > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(28) Obligation:

Q DP problem:
P is empty.
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.

(29) PisEmptyProof (EQUIVALENT transformation)

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

(30) TRUE

(31) Obligation:

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))

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U641(ok(X1), ok(X2), ok(X3)) → U641(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U641(x1, x2, x3)  =  U641(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  x2
tt  =  tt
U12(x1, x2, x3)  =  x3
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U64^11 > [mark, active1]
proper > tt > [ok1, 0] > [mark, active1]
top > [mark, active1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(33) Obligation:

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

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))

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U641(mark(X1), X2, X3) → U641(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U641(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U113, U623] > U123 > U133 > [isNatKind, s1, plus2] > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U143 > U152 > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U212 > [isNatKind, s1, plus2] > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U212 > U222 > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U321 > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U512 > [isNatKind, s1, plus2] > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U512 > U522 > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U613 > [isNatKind, s1, plus2] > [mark1, tt, isNat1, U161, U411]
[active1, U113, U623] > U633 > U643 > [isNatKind, s1, plus2] > [mark1, tt, isNat1, U161, U411]
0 > U512 > [isNatKind, s1, plus2] > [mark1, tt, isNat1, U161, U411]
0 > U512 > U522 > [mark1, tt, isNat1, U161, U411]
top > [mark1, tt, isNat1, U161, U411]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(35) Obligation:

Q DP problem:
P is empty.
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.

(36) PisEmptyProof (EQUIVALENT transformation)

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

(37) TRUE

(38) Obligation:

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

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

The TRS R consists of the following rules:

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U631(mark(X1), X2, X3) → U631(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U631(x1, x2, x3)  =  U631(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[proper1, top] > [active1, U613, U623, U643] > U151 > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > [active1, U613, U623, U643] > U211 > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > [active1, U613, U623, U643] > U231 > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > [active1, U613, U623, U643] > U411 > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > [active1, U613, U623, U643] > [U512, plus2] > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > [active1, U613, U623, U643] > U522 > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > [active1, U613, U623, U643] > U633 > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]
[proper1, top] > 0 > [U512, plus2] > [U63^11, mark1, U111, U131, U321] > [tt, isNatKind, isNat]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(40) Obligation:

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

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))

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U631(ok(X1), ok(X2), ok(X3)) → U631(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U631(x1, x2, x3)  =  U631(x3)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U11(x1, x2, x3)  =  x1
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  U62(x3)
U63(x1, x2, x3)  =  U63(x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
0 > [tt, mark, U121, U131, U141, U161, U211, U511, U621, U631, U642, s1, plus1, proper1] > U231 > [U63^11, ok1, U151, isNat1]
0 > [tt, mark, U121, U131, U141, U161, U211, U511, U621, U631, U642, s1, plus1, proper1] > U522 > [U63^11, ok1, U151, isNat1]
top > [tt, mark, U121, U131, U141, U161, U211, U511, U621, U631, U642, s1, plus1, proper1] > U231 > [U63^11, ok1, U151, isNat1]
top > [tt, mark, U121, U131, U141, U161, U211, U511, U621, U631, U642, s1, plus1, proper1] > U522 > [U63^11, ok1, U151, isNat1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(42) Obligation:

Q DP problem:
P is empty.
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.

(43) PisEmptyProof (EQUIVALENT transformation)

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

(44) TRUE

(45) Obligation:

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))

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

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(ok(X1), ok(X2), ok(X3)) → U621(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2, x3)  =  U621(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  x1
U11(x1, x2, x3)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  x3
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x1
isNat(x1)  =  x1
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x2)
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  plus(x1)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[U161, U221, U311, proper1] > U321 > [U62^11, ok1] > top > [mark, 0]
[U161, U221, U311, proper1] > U321 > tt > [mark, 0]
[U161, U221, U311, proper1] > U511 > [U62^11, ok1] > top > [mark, 0]
[U161, U221, U311, proper1] > plus1 > [U62^11, ok1] > top > [mark, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(47) Obligation:

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

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))

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

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(mark(X1), X2, X3) → U621(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2, x3)  =  U621(x1, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U62^12 > [isNat, top]
0 > [active1, tt, U161, plus2] > U151 > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > U211 > [U111, isNatKind] > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > U231 > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > U512 > [U111, isNatKind] > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > U512 > U522 > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > U633 > [U111, isNatKind] > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > U633 > U643 > [mark1, U121, U321] > [isNat, top]
0 > [active1, tt, U161, plus2] > s1 > [U613, U623] > [U111, isNatKind] > [mark1, U121, U321] > [isNat, top]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(49) Obligation:

Q DP problem:
P is empty.
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.

(50) PisEmptyProof (EQUIVALENT transformation)

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

(51) TRUE

(52) Obligation:

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

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

The TRS R consists of the following rules:

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2, x3)  =  U611(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  x1
U11(x1, x2, x3)  =  x3
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x1
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  x2
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  x2
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U61^11 > mark
proper > ok1 > mark
proper > tt > mark
proper > 0 > mark
top > mark


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(54) Obligation:

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

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

The TRS R consists of the following rules:

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

(55) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X1), X2, X3) → U611(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2, x3)  =  U611(x1, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U61^12 > top
0 > [tt, isNatKind, U211, U512, U643] > [isNat, plus2] > [mark1, U111, U141, U321] > top
proper1 > [active1, U221, U613, U623] > U121 > U131 > [tt, isNatKind, U211, U512, U643] > [isNat, plus2] > [mark1, U111, U141, U321] > top
proper1 > [active1, U221, U613, U623] > U522 > [mark1, U111, U141, U321] > top
proper1 > [active1, U221, U613, U623] > U633 > [tt, isNatKind, U211, U512, U643] > [isNat, plus2] > [mark1, U111, U141, U321] > top


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(56) Obligation:

Q DP problem:
P is empty.
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.

(57) PisEmptyProof (EQUIVALENT transformation)

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

(58) TRUE

(59) Obligation:

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

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

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(mark(X1), X2) → U521(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U521(x1, x2)  =  U521(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > U321 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > U613 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > s1 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > plus2 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(61) Obligation:

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

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))

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(ok(X1), ok(X2)) → U521(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U521(x1, x2)  =  U521(x1)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U11(x1, x2, x3)  =  x1
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x2
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x3
s(x1)  =  x1
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > proper > [U52^11, ok1] > [tt, mark, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(63) Obligation:

Q DP problem:
P is empty.
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.

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) Obligation:

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))

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(mark(X1), X2) → U511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  U511(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > U321 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > U613 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > s1 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]
top > [active1, U211, U512, U522, U623, U633, U643, proper1] > plus2 > [mark1, U111, U121, U131, U151, U161, U221, U411] > [isNat, 0] > [tt, isNatKind]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(68) Obligation:

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

U511(ok(X1), ok(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))

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(ok(X1), ok(X2)) → U511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  U511(x1)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U11(x1, x2, x3)  =  x1
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x2
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x3
s(x1)  =  x1
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > proper > [U51^11, ok1] > [tt, mark, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(70) Obligation:

Q DP problem:
P is empty.
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.

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) Obligation:

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))

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(ok(X)) → U411(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1)  =  U411(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U141 > [ok1, U111, U121, U131, U221, U321] > U41^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U151 > [ok1, U111, U121, U131, U221, U321] > U41^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > [U211, s1] > U613 > [ok1, U111, U121, U131, U221, U321] > U41^11


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(75) Obligation:

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

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))

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(mark(X)) → U411(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U512, plus2, top] > U121 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U161 > tt > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U522 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U613 > U623 > U633 > U643 > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(77) Obligation:

Q DP problem:
P is empty.
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.

(78) PisEmptyProof (EQUIVALENT transformation)

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

(79) TRUE

(80) Obligation:

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))

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(ok(X)) → U321(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1)  =  U321(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U141 > [ok1, U111, U121, U131, U221, U321] > U32^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U151 > [ok1, U111, U121, U131, U221, U321] > U32^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > [U211, s1] > U613 > [ok1, U111, U121, U131, U221, U321] > U32^11


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(82) Obligation:

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

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))

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

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(mark(X)) → U321(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U512, plus2, top] > U121 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U161 > tt > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U522 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U613 > U623 > U633 > U643 > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(84) Obligation:

Q DP problem:
P is empty.
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.

(85) PisEmptyProof (EQUIVALENT transformation)

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

(86) TRUE

(87) Obligation:

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

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

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(mark(X1), X2) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2)  =  U311(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x1)
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U31^12 > [tt, isNatKind, isNat]
0 > [mark1, U111, U121, U131, U211, U221, U231] > [tt, isNatKind, isNat]
top > [active1, U321, U512, U623, U643] > U161 > [mark1, U111, U121, U131, U211, U221, U231] > [tt, isNatKind, isNat]
top > [active1, U321, U512, U623, U643] > U522 > [mark1, U111, U121, U131, U211, U221, U231] > [tt, isNatKind, isNat]
top > [active1, U321, U512, U623, U643] > [U613, plus2] > U311 > [mark1, U111, U121, U131, U211, U221, U231] > [tt, isNatKind, isNat]
top > [active1, U321, U512, U623, U643] > U633 > [mark1, U111, U121, U131, U211, U221, U231] > [tt, isNatKind, isNat]
top > [active1, U321, U512, U623, U643] > s1 > U411 > [mark1, U111, U121, U131, U211, U221, U231] > [tt, isNatKind, isNat]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(89) Obligation:

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

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))

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

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(ok(X1), ok(X2)) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2)  =  x1
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x2)
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  U12(x2)
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  x1
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > proper1 > [active1, U111, U121, U411] > U131 > [ok1, U141, U151, U211, U221] > [tt, mark, 0]
top > proper1 > [active1, U111, U121, U411] > [U612, U622, U632, U642, s1] > [U512, plus2] > [ok1, U141, U151, U211, U221] > [tt, mark, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(91) Obligation:

Q DP problem:
P is empty.
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.

(92) PisEmptyProof (EQUIVALENT transformation)

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

(93) TRUE

(94) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U231(ok(X)) → U231(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U231(x1)  =  U231(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U141 > [ok1, U111, U121, U131, U221, U321] > U23^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U151 > [ok1, U111, U121, U131, U221, U321] > U23^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > [U211, s1] > U613 > [ok1, U111, U121, U131, U221, U321] > U23^11


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(96) Obligation:

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

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))

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

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U231(mark(X)) → U231(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U231(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U512, plus2, top] > U121 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U161 > tt > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U522 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U613 > U623 > U633 > U643 > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(98) Obligation:

Q DP problem:
P is empty.
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.

(99) PisEmptyProof (EQUIVALENT transformation)

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

(100) TRUE

(101) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(ok(X1), ok(X2)) → U221(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1, x2)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, tt, proper1, top] > [U112, isNat1] > U123 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U231 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U312 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U612 > U622 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U632 > U642 > s1 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U632 > U642 > plus2 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > 0 > [ok1, U141, U151, U161, U321, U411, U521]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(103) Obligation:

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

U221(mark(X1), 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))

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

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(mark(X1), X2) → U221(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x1)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U111 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U141 > U151 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U231 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U321 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U522 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U623 > U633 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > plus2 > U613 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U211 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U111 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U141 > U151 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U231 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U321 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U522 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U623 > U633 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > plus2 > U613 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U211 > mark1
top > mark1


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(105) Obligation:

Q DP problem:
P is empty.
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.

(106) PisEmptyProof (EQUIVALENT transformation)

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

(107) TRUE

(108) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(ok(X1), ok(X2)) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  x2
U22(x1, x2)  =  x2
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, tt, proper1, top] > [U112, isNat1] > U123 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U231 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U312 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U612 > U622 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U632 > U642 > s1 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > U632 > U642 > plus2 > [ok1, U141, U151, U161, U321, U411, U521]
[active1, tt, proper1, top] > 0 > [ok1, U141, U151, U161, U321, U411, U521]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(110) Obligation:

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

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))

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

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(mark(X1), X2) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x1)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U111 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U141 > U151 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U231 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U321 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U522 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U623 > U633 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > plus2 > U613 > mark1
active1 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U211 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U111 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U141 > U151 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U231 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U321 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U522 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > U623 > U633 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > tt > plus2 > U613 > mark1
0 > [U121, isNatKind, isNat, U311, U512, U643, s1] > U211 > mark1
top > mark1


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(112) Obligation:

Q DP problem:
P is empty.
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.

(113) PisEmptyProof (EQUIVALENT transformation)

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

(114) TRUE

(115) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U161(ok(X)) → U161(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U161(x1)  =  U161(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x3)
U14(x1, x2, x3)  =  U14(x3)
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  x1
U31(x1, x2)  =  x2
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  U52(x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U141 > [ok1, U111, U121, U131, U221, U321] > U16^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > U151 > [ok1, U111, U121, U131, U221, U321] > U16^11
[active1, isNat1, U521, U632, proper1] > isNatKind1 > [tt, U623, U642, plus2, 0] > [U211, s1] > U613 > [ok1, U111, U121, U131, U221, U321] > U16^11


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(117) Obligation:

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

U161(mark(X)) → U161(X)

The TRS R consists of the following rules:

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

(118) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U161(mark(X)) → U161(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U161(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U512, plus2, top] > U121 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U161 > tt > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U522 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]
[active1, U512, plus2, top] > U613 > U623 > U633 > U643 > s1 > [mark1, U111, U131, U141, U151, U211, U221, U311] > isNatKind > [isNat, 0, ok]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(119) Obligation:

Q DP problem:
P is empty.
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.

(120) PisEmptyProof (EQUIVALENT transformation)

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

(121) TRUE

(122) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U151(mark(X1), X2) → U151(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151(x1, x2)  =  U151(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U15^11 > [tt, isNatKind, isNat, 0]
[active1, U131, U141, U512, U522, U613, U623, U633, U643, plus2] > [mark1, U121, U161, U211, U221, U321, s1] > top > [tt, isNatKind, isNat, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(124) Obligation:

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

U151(ok(X1), ok(X2)) → U151(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(125) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U151(ok(X1), ok(X2)) → U151(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U151(x1, x2)  =  U151(x1)
ok(x1)  =  ok(x1)
active(x1)  =  x1
U11(x1, x2, x3)  =  x3
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  x3
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x2
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x2
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  x3
s(x1)  =  x1
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
top > proper > [U15^11, ok1] > [tt, mark, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(126) Obligation:

Q DP problem:
P is empty.
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.

(127) PisEmptyProof (EQUIVALENT transformation)

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

(128) TRUE

(129) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(130) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U141(ok(X1), ok(X2), ok(X3)) → U141(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141(x1, x2, x3)  =  U141(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  x2
tt  =  tt
U12(x1, x2, x3)  =  x3
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U14^11 > [mark, active1]
proper > tt > [ok1, 0] > [mark, active1]
top > [mark, active1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(131) Obligation:

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

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))

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

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U141(mark(X1), X2, X3) → U141(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U141(x1, x2, x3)  =  U141(x1, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > isNatKind > [mark1, U111, U161, U231, U321, 0] > U14^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > U131 > [mark1, U111, U161, U231, U321, 0] > U14^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > U522 > [mark1, U111, U161, U231, U321, 0] > U14^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > U613 > [mark1, U111, U161, U231, U321, 0] > U14^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > [U633, U643] > [mark1, U111, U161, U231, U321, 0] > U14^12 > top


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(133) Obligation:

Q DP problem:
P is empty.
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.

(134) PisEmptyProof (EQUIVALENT transformation)

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

(135) TRUE

(136) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U131(ok(X1), ok(X2), ok(X3)) → U131(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131(x1, x2, x3)  =  U131(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  x2
tt  =  tt
U12(x1, x2, x3)  =  x3
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x2
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x2
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2, x3)  =  x3
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U13^11 > [mark, active1]
proper > tt > [ok1, 0] > [mark, active1]
top > [mark, active1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(138) Obligation:

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

U131(mark(X1), X2, 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))

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

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U131(mark(X1), X2, X3) → U131(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131(x1, x2, x3)  =  U131(x1, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > isNatKind > [mark1, U111, U161, U231, U321, 0] > U13^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > U131 > [mark1, U111, U161, U231, U321, 0] > U13^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > U522 > [mark1, U111, U161, U231, U321, 0] > U13^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > U613 > [mark1, U111, U161, U231, U321, 0] > U13^12 > top
active1 > [U121, isNat, U211, U512, U623, plus2, proper1] > [tt, U151, s1] > [U633, U643] > [mark1, U111, U161, U231, U321, 0] > U13^12 > top


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(140) Obligation:

Q DP problem:
P is empty.
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.

(141) PisEmptyProof (EQUIVALENT transformation)

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

(142) TRUE

(143) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(ok(X1), ok(X2), ok(X3)) → U121(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2, x3)  =  U121(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark
active(x1)  =  x1
U11(x1, x2, x3)  =  x3
tt  =  tt
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  x1
isNat(x1)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  x2
U63(x1, x2, x3)  =  x3
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  x2
0  =  0
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U12^11 > mark
proper > ok1 > mark
proper > tt > mark
proper > 0 > mark
top > mark


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(145) Obligation:

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

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))

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

(146) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(mark(X1), X2, X3) → U121(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1, x2, x3)  =  U121(x1, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
isNat(x1)  =  isNat
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U12^12 > top
0 > [tt, isNatKind, U211, U512, U643] > [isNat, plus2] > [mark1, U111, U141, U321] > top
proper1 > [active1, U221, U613, U623] > U121 > U131 > [tt, isNatKind, U211, U512, U643] > [isNat, plus2] > [mark1, U111, U141, U321] > top
proper1 > [active1, U221, U613, U623] > U522 > [mark1, U111, U141, U321] > top
proper1 > [active1, U221, U613, U623] > U633 > [tt, isNatKind, U211, U512, U643] > [isNat, plus2] > [mark1, U111, U141, U321] > top


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(147) Obligation:

Q DP problem:
P is empty.
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.

(148) PisEmptyProof (EQUIVALENT transformation)

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

(149) TRUE

(150) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.

(151) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(ok(X1), ok(X2), ok(X3)) → U111(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  x2
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x2, x3)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1, x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  x3
U14(x1, x2, x3)  =  x3
U15(x1, x2)  =  U15(x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x2)
U22(x1, x2)  =  U22(x2)
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x2)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x2, x3)
U64(x1, x2, x3)  =  U64(x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, isNatKind1, U612, U623, U632, plus2, proper1] > [U112, U122] > [ok1, tt, U151, isNat1, U211, U221, U311, U321, U511, top]
[active1, isNatKind1, U612, U623, U632, plus2, proper1] > U231 > [ok1, tt, U151, isNat1, U211, U221, U311, U321, U511, top]
[active1, isNatKind1, U612, U623, U632, plus2, proper1] > U522 > [ok1, tt, U151, isNat1, U211, U221, U311, U321, U511, top]
[active1, isNatKind1, U612, U623, U632, plus2, proper1] > U642 > [ok1, tt, U151, isNat1, U211, U221, U311, U321, U511, top]
0 > [ok1, tt, U151, isNat1, U211, U221, U311, U321, U511, top]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(152) Obligation:

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

U111(mark(X1), X2, 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))

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

(153) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(mark(X1), X2, X3) → U111(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2, x3)  =  U111(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12(x1)
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  x1
U23(x1)  =  U23(x1)
U31(x1, x2)  =  U31(x1)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U633, U643] > U321 > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > U512 > isNatKind > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > U512 > isNatKind > U311 > [U11^11, mark1, U161]
[active1, U633, U643] > U512 > U522 > [U11^11, mark1, U161]
[active1, U633, U643] > U623 > isNat > U111 > U121 > U131 > isNatKind > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > U623 > isNat > U111 > U121 > U131 > isNatKind > U311 > [U11^11, mark1, U161]
[active1, U633, U643] > U623 > isNat > U111 > U121 > U131 > U141 > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > U623 > isNat > U211 > isNatKind > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > U623 > isNat > U211 > isNatKind > U311 > [U11^11, mark1, U161]
[active1, U633, U643] > s1 > U211 > isNatKind > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > s1 > U211 > isNatKind > U311 > [U11^11, mark1, U161]
[active1, U633, U643] > s1 > U613 > [U11^11, mark1, U161]
[active1, U633, U643] > plus2 > isNatKind > [tt, U151] > U231 > [U11^11, mark1, U161]
[active1, U633, U643] > plus2 > isNatKind > U311 > [U11^11, mark1, U161]
[active1, U633, U643] > plus2 > U613 > [U11^11, mark1, U161]
0 > [tt, U151] > U231 > [U11^11, mark1, U161]
top > [U11^11, mark1, U161]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(154) Obligation:

Q DP problem:
P is empty.
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.

(155) PisEmptyProof (EQUIVALENT transformation)

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

(156) TRUE

(157) Obligation:

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

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

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U11(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2, X3)) → PROPER(X1)
PROPER(U11(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X1, X2, X3)) → PROPER(X2)
PROPER(U12(X1, X2, X3)) → PROPER(X3)
PROPER(isNatKind(X)) → PROPER(X)
PROPER(U13(X1, X2, X3)) → PROPER(X1)
PROPER(U13(X1, X2, X3)) → PROPER(X2)
PROPER(U13(X1, X2, X3)) → PROPER(X3)
PROPER(U14(X1, X2, X3)) → PROPER(X1)
PROPER(U14(X1, X2, X3)) → PROPER(X2)
PROPER(U14(X1, X2, X3)) → PROPER(X3)
PROPER(U15(X1, X2)) → PROPER(X1)
PROPER(U15(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(U16(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X1, X2)) → PROPER(X1)
PROPER(U22(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2, X3)) → PROPER(X1)
PROPER(U62(X1, X2, X3)) → PROPER(X2)
PROPER(U62(X1, X2, X3)) → PROPER(X3)
PROPER(U63(X1, X2, X3)) → PROPER(X1)
PROPER(U63(X1, X2, X3)) → PROPER(X2)
PROPER(U63(X1, X2, X3)) → PROPER(X3)
PROPER(U64(X1, X2, X3)) → PROPER(X1)
PROPER(U64(X1, X2, X3)) → PROPER(X2)
PROPER(U64(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U11(x1, x2, x3)  =  U11(x1, x2, x3)
U12(x1, x2, x3)  =  U12(x1, x2, x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x1, x2, x3)
U14(x1, x2, x3)  =  U14(x1, x2, x3)
U15(x1, x2)  =  U15(x1, x2)
isNat(x1)  =  isNat(x1)
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  x1
U41(x1)  =  U41(x1)
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
PROPER1 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U143 > isNat1 > [isNatKind1, U152, tt] > U133 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U143 > isNat1 > [isNatKind1, U152, tt] > U411 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U143 > isNat1 > [isNatKind1, U152, tt] > U623 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U143 > isNat1 > U212 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U222 > isNat1 > [isNatKind1, U152, tt] > U133 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U222 > isNat1 > [isNatKind1, U152, tt] > U411 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U222 > isNat1 > [isNatKind1, U152, tt] > U623 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U222 > isNat1 > U212 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > [U613, plus2] > isNat1 > [isNatKind1, U152, tt] > U133 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > [U613, plus2] > isNat1 > [isNatKind1, U152, tt] > U411 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > [U613, plus2] > isNat1 > [isNatKind1, U152, tt] > U623 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > [U613, plus2] > isNat1 > U212 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U633 > [isNatKind1, U152, tt] > U133 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U633 > [isNatKind1, U152, tt] > U411 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > U633 > [isNatKind1, U152, tt] > U623 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > 0 > isNat1 > [isNatKind1, U152, tt] > U133 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > 0 > isNat1 > [isNatKind1, U152, tt] > U411 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > 0 > isNat1 > [isNatKind1, U152, tt] > U623 > [U123, s1]
top > active1 > [U113, U161, U312, U512, U522, U643, proper1] > 0 > isNat1 > U212 > [U123, s1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(159) Obligation:

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

PROPER(U23(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(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.

(160) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U23(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
U23(x1)  =  U23(x1)
U32(x1)  =  U32(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
mark(x1)  =  mark(x1)
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U16(x1)  =  U16(x1)
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U31(x1, x2)  =  U31(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
0 > [active1, U512] > U231 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U231 > tt > isNatKind
0 > [active1, U512] > U321 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U321 > tt > isNatKind
0 > [active1, U512] > U161 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U161 > tt > isNatKind
0 > [active1, U512] > U522 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U643 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > plus2 > [isNat, U613, U623] > tt > isNatKind
0 > [active1, U512] > plus2 > [isNat, U613, U623] > U633 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U231 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U231 > tt > isNatKind
top > [active1, U512] > U321 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U321 > tt > isNatKind
top > [active1, U512] > U161 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U161 > tt > isNatKind
top > [active1, U512] > U522 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U643 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > plus2 > [isNat, U613, U623] > tt > isNatKind
top > [active1, U512] > plus2 > [isNat, U613, U623] > U633 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(161) Obligation:

Q DP problem:
P is empty.
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.

(162) PisEmptyProof (EQUIVALENT transformation)

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

(163) TRUE

(164) Obligation:

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

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

(165) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U12(x1, x2, x3)  =  x1
U11(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  x1
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  mark
isNatKind(x1)  =  isNatKind
isNat(x1)  =  isNat
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, tt] > plus2 > [mark, isNatKind, top]
[active1, tt] > plus2 > isNat
0 > isNat


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(166) Obligation:

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

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

(167) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(s(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U12(x1, x2, x3)  =  x1
U11(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  x1
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  x1
s(x1)  =  s(x1)
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  mark
isNatKind(x1)  =  isNatKind(x1)
isNat(x1)  =  isNat(x1)
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  ok(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
proper1 > isNatKind1 > [ACTIVE1, s1, active1, isNat1, ok1] > top > [tt, mark]
proper1 > 0 > [tt, mark]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(168) Obligation:

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

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

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

(169) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U12(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U13(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U14(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U62(X1, X2, X3)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U12(x1, x2, x3)  =  U12(x1)
U11(x1, x2, x3)  =  x1
U13(x1, x2, x3)  =  U13(x1)
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  x1
U16(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1, x2)  =  U22(x1)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  x1
U62(x1, x2, x3)  =  U62(x1)
U63(x1, x2, x3)  =  x1
U64(x1, x2, x3)  =  x1
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  x1
isNatKind(x1)  =  isNatKind
isNat(x1)  =  isNat
s(x1)  =  s
plus(x1, x2)  =  x1
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
0 > [tt, isNatKind] > [U512, isNat]
top > [U121, U131, U221, U311, active1] > U141 > [U512, isNat]
top > [U121, U131, U221, U311, active1] > U522 > [U512, isNat]
top > [U121, U131, U221, U311, active1] > U621 > [U512, isNat]
top > [U121, U131, U221, U311, active1] > s > [tt, isNatKind] > [U512, isNat]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(170) Obligation:

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

ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U15(X1, X2)) → ACTIVE(X1)
ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U23(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U63(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U64(X1, X2, X3)) → 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))

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

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U11(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U15(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U63(X1, X2, X3)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1, x2, x3)  =  U11(x1, x3)
U15(x1, x2)  =  U15(x1, x2)
U16(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U23(x1)  =  x1
U32(x1)  =  U32(x1)
U41(x1)  =  U41(x1)
U61(x1, x2, x3)  =  U61(x1)
U63(x1, x2, x3)  =  U63(x1, x3)
U64(x1, x2, x3)  =  x1
active(x1)  =  x1
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  U12(x1, x3)
isNatKind(x1)  =  isNatKind(x1)
U13(x1, x2, x3)  =  U13(x2, x3)
U14(x1, x2, x3)  =  U14(x2)
isNat(x1)  =  x1
U22(x1, x2)  =  U22(x1)
U31(x1, x2)  =  U31(x2)
U51(x1, x2)  =  x2
U52(x1, x2)  =  x2
U62(x1, x2, x3)  =  U62
s(x1)  =  s(x1)
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ACTIVE1 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U112 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U152 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U212 > U221 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U321 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U611 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U122 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > isNatKind1 > U411 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U132 > U141 > [mark, U311, s1, plus2, 0]
tt > [U632, proper1] > U62 > [mark, U311, s1, plus2, 0]
top > [mark, U311, s1, plus2, 0]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(172) Obligation:

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

ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U23(X)) → ACTIVE(X)
ACTIVE(U64(X1, X2, X3)) → 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))

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

(173) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U64(X1, X2, X3)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U16(x1)  =  x1
U23(x1)  =  x1
U64(x1, x2, x3)  =  U64(x1)
active(x1)  =  x1
U11(x1, x2, x3)  =  U11
tt  =  tt
mark(x1)  =  mark
U12(x1, x2, x3)  =  U12
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  U13
U14(x1, x2, x3)  =  x1
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22(x1)
U31(x1, x2)  =  U31
U32(x1)  =  x1
U41(x1)  =  U41
U51(x1, x2)  =  U51
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  U61
U62(x1, x2, x3)  =  x1
U63(x1, x2, x3)  =  U63
s(x1)  =  x1
plus(x1, x2)  =  plus
0  =  0
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U12 > [U641, U11, tt, mark, isNatKind, U13, U151, isNat1, U221, U41, U51, U61, U63, 0, proper1]
U21 > [U641, U11, tt, mark, isNatKind, U13, U151, isNat1, U221, U41, U51, U61, U63, 0, proper1]
U31 > [U641, U11, tt, mark, isNatKind, U13, U151, isNat1, U221, U41, U51, U61, U63, 0, proper1]
plus > [U641, U11, tt, mark, isNatKind, U13, U151, isNat1, U221, U41, U51, U61, U63, 0, proper1]
top > [U641, U11, tt, mark, isNatKind, U13, U151, isNat1, U221, U41, U51, U61, U63, 0, proper1]


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(174) Obligation:

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

ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U23(X)) → 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.

(175) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U16(X)) → ACTIVE(X)
ACTIVE(U23(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U16(x1)  =  U16(x1)
U23(x1)  =  U23(x1)
active(x1)  =  active(x1)
U11(x1, x2, x3)  =  U11(x1)
tt  =  tt
mark(x1)  =  mark(x1)
U12(x1, x2, x3)  =  x1
isNatKind(x1)  =  isNatKind
U13(x1, x2, x3)  =  x1
U14(x1, x2, x3)  =  U14(x1)
U15(x1, x2)  =  U15(x1)
isNat(x1)  =  isNat
U21(x1, x2)  =  U21(x1)
U22(x1, x2)  =  U22(x1)
U31(x1, x2)  =  U31(x1)
U32(x1)  =  U32(x1)
U41(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2, x3)  =  U62(x1, x2, x3)
U63(x1, x2, x3)  =  U63(x1, x2, x3)
U64(x1, x2, x3)  =  U64(x1, x2, x3)
s(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
0  =  0
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
0 > [active1, U512] > U161 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U161 > tt > isNatKind
0 > [active1, U512] > U231 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U231 > tt > isNatKind
0 > [active1, U512] > U321 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U321 > tt > isNatKind
0 > [active1, U512] > U522 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > U643 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
0 > [active1, U512] > plus2 > [isNat, U613, U623] > tt > isNatKind
0 > [active1, U512] > plus2 > [isNat, U613, U623] > U633 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U161 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U161 > tt > isNatKind
top > [active1, U512] > U231 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U231 > tt > isNatKind
top > [active1, U512] > U321 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U321 > tt > isNatKind
top > [active1, U512] > U522 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > U643 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind
top > [active1, U512] > plus2 > [isNat, U613, U623] > tt > isNatKind
top > [active1, U512] > plus2 > [isNat, U613, U623] > U633 > [U111, mark1, U141, U151, U211, U221, U311] > isNatKind


The following usable rules [FROCOS05] were oriented:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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))

(176) Obligation:

Q DP problem:
P is empty.
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.

(177) PisEmptyProof (EQUIVALENT transformation)

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

(178) TRUE

(179) Obligation:

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

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

The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(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.