Termination w.r.t. Q of the given QTRS could be disproven:

(0) Obligation:

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

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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:

ZEROS → CONS(0, n__zeros)
ZEROS → 0¹
U101¹(tt, V1, V2) → U102¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U101¹(tt, V1, V2) → ISNATKIND(activate(V1))
U101¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ACTIVATE(V2)
U102¹(tt, V1, V2) → U103¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U102¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U102¹(tt, V1, V2) → ACTIVATE(V2)
U102¹(tt, V1, V2) → ACTIVATE(V1)
U103¹(tt, V1, V2) → U104¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U103¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ACTIVATE(V1)
U104¹(tt, V1, V2) → U105¹(isNat(activate(V1)), activate(V2))
U104¹(tt, V1, V2) → ISNAT(activate(V1))
U104¹(tt, V1, V2) → ACTIVATE(V1)
U104¹(tt, V1, V2) → ACTIVATE(V2)
U105¹(tt, V2) → U106¹(isNatIList(activate(V2)))
U105¹(tt, V2) → ISNATILIST(activate(V2))
U105¹(tt, V2) → ACTIVATE(V2)
U11¹(tt, V1) → U12¹(isNatIListKind(activate(V1)), activate(V1))
U11¹(tt, V1) → ISNATILISTKIND(activate(V1))
U11¹(tt, V1) → ACTIVATE(V1)
U111¹(tt, L, N) → U112¹(isNatIListKind(activate(L)), activate(L), activate(N))
U111¹(tt, L, N) → ISNATILISTKIND(activate(L))
U111¹(tt, L, N) → ACTIVATE(L)
U111¹(tt, L, N) → ACTIVATE(N)
U112¹(tt, L, N) → U113¹(isNat(activate(N)), activate(L), activate(N))
U112¹(tt, L, N) → ISNAT(activate(N))
U112¹(tt, L, N) → ACTIVATE(N)
U112¹(tt, L, N) → ACTIVATE(L)
U113¹(tt, L, N) → U114¹(isNatKind(activate(N)), activate(L))
U113¹(tt, L, N) → ISNATKIND(activate(N))
U113¹(tt, L, N) → ACTIVATE(N)
U113¹(tt, L, N) → ACTIVATE(L)
U114¹(tt, L) → S(length(activate(L)))
U114¹(tt, L) → LENGTH(activate(L))
U114¹(tt, L) → ACTIVATE(L)
U12¹(tt, V1) → U13¹(isNatList(activate(V1)))
U12¹(tt, V1) → ISNATLIST(activate(V1))
U12¹(tt, V1) → ACTIVATE(V1)
U121¹(tt, IL) → U122¹(isNatIListKind(activate(IL)))
U121¹(tt, IL) → ISNATILISTKIND(activate(IL))
U121¹(tt, IL) → ACTIVATE(IL)
U122¹(tt) → NIL
U131¹(tt, IL, M, N) → U132¹(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U131¹(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U131¹(tt, IL, M, N) → ACTIVATE(IL)
U131¹(tt, IL, M, N) → ACTIVATE(M)
U131¹(tt, IL, M, N) → ACTIVATE(N)
U132¹(tt, IL, M, N) → U133¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U132¹(tt, IL, M, N) → ISNAT(activate(M))
U132¹(tt, IL, M, N) → ACTIVATE(M)
U132¹(tt, IL, M, N) → ACTIVATE(IL)
U132¹(tt, IL, M, N) → ACTIVATE(N)
U133¹(tt, IL, M, N) → U134¹(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U133¹(tt, IL, M, N) → ISNATKIND(activate(M))
U133¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ACTIVATE(IL)
U133¹(tt, IL, M, N) → ACTIVATE(N)
U134¹(tt, IL, M, N) → U135¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U134¹(tt, IL, M, N) → ISNAT(activate(N))
U134¹(tt, IL, M, N) → ACTIVATE(N)
U134¹(tt, IL, M, N) → ACTIVATE(IL)
U134¹(tt, IL, M, N) → ACTIVATE(M)
U135¹(tt, IL, M, N) → U136¹(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U135¹(tt, IL, M, N) → ISNATKIND(activate(N))
U135¹(tt, IL, M, N) → ACTIVATE(N)
U135¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ACTIVATE(M)
U136¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
U136¹(tt, IL, M, N) → ACTIVATE(N)
U136¹(tt, IL, M, N) → ACTIVATE(M)
U136¹(tt, IL, M, N) → ACTIVATE(IL)
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U22¹(tt, V1) → U23¹(isNat(activate(V1)))
U22¹(tt, V1) → ISNAT(activate(V1))
U22¹(tt, V1) → ACTIVATE(V1)
U31¹(tt, V) → U32¹(isNatIListKind(activate(V)), activate(V))
U31¹(tt, V) → ISNATILISTKIND(activate(V))
U31¹(tt, V) → ACTIVATE(V)
U32¹(tt, V) → U33¹(isNatList(activate(V)))
U32¹(tt, V) → ISNATLIST(activate(V))
U32¹(tt, V) → ACTIVATE(V)
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U42¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → ACTIVATE(V1)
U43¹(tt, V1, V2) → U44¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U43¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → U45¹(isNat(activate(V1)), activate(V2))
U44¹(tt, V1, V2) → ISNAT(activate(V1))
U44¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V2)
U45¹(tt, V2) → U46¹(isNatIList(activate(V2)))
U45¹(tt, V2) → ISNATILIST(activate(V2))
U45¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, V2) → U52¹(isNatIListKind(activate(V2)))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
U51¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V2) → U62¹(isNatIListKind(activate(V2)))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))
U61¹(tt, V2) → ACTIVATE(V2)
U91¹(tt, V1, V2) → U92¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U91¹(tt, V1, V2) → ISNATKIND(activate(V1))
U91¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ACTIVATE(V2)
U92¹(tt, V1, V2) → U93¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U92¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U92¹(tt, V1, V2) → ACTIVATE(V2)
U92¹(tt, V1, V2) → ACTIVATE(V1)
U93¹(tt, V1, V2) → U94¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U93¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ACTIVATE(V1)
U94¹(tt, V1, V2) → U95¹(isNat(activate(V1)), activate(V2))
U94¹(tt, V1, V2) → ISNAT(activate(V1))
U94¹(tt, V1, V2) → ACTIVATE(V1)
U94¹(tt, V1, V2) → ACTIVATE(V2)
U95¹(tt, V2) → U96¹(isNatList(activate(V2)))
U95¹(tt, V2) → ISNATLIST(activate(V2))
U95¹(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__length(V1)) → U71¹(isNatIListKind(activate(V1)))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → U81¹(isNatKind(activate(V1)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → U91¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U101¹(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(nil) → 0¹
LENGTH(cons(N, L)) → U111¹(isNatList(activate(L)), activate(L), N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(0, IL) → U121¹(isNatIList(IL), IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U131¹(isNatIList(activate(IL)), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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 1 SCC with 22 less nodes.

(4) Obligation:

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

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → U121¹(isNatIList(IL), IL)
U121¹(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → U111¹(isNatList(activate(L)), activate(L), N)
U111¹(tt, L, N) → U112¹(isNatIListKind(activate(L)), activate(L), activate(N))
U112¹(tt, L, N) → U113¹(isNat(activate(N)), activate(L), activate(N))
U113¹(tt, L, N) → U114¹(isNatKind(activate(N)), activate(L))
U114¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U91¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U91¹(tt, V1, V2) → U92¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U92¹(tt, V1, V2) → U93¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93¹(tt, V1, V2) → U94¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94¹(tt, V1, V2) → U95¹(isNat(activate(V1)), activate(V2))
U95¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U101¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U101¹(tt, V1, V2) → U102¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U102¹(tt, V1, V2) → U103¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103¹(tt, V1, V2) → U104¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104¹(tt, V1, V2) → U105¹(isNat(activate(V1)), activate(V2))
U105¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U31¹(tt, V) → U32¹(isNatIListKind(activate(V)), activate(V))
U32¹(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U32¹(tt, V) → ACTIVATE(V)
U31¹(tt, V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, V) → ACTIVATE(V)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isNat(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U45¹(tt, V2) → ACTIVATE(V2)
U44¹(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
U11¹(tt, V1) → U12¹(isNatIListKind(activate(V1)), activate(V1))
U12¹(tt, V1) → ISNATLIST(activate(V1))
U12¹(tt, V1) → ACTIVATE(V1)
U11¹(tt, V1) → ISNATILISTKIND(activate(V1))
U11¹(tt, V1) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U43¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ACTIVATE(V1)
U42¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U42¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U105¹(tt, V2) → ACTIVATE(V2)
U104¹(tt, V1, V2) → ISNAT(activate(V1))
U104¹(tt, V1, V2) → ACTIVATE(V1)
U104¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U103¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ACTIVATE(V1)
U102¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U102¹(tt, V1, V2) → ACTIVATE(V2)
U102¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ISNATKIND(activate(V1))
U101¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ACTIVATE(V2)
U95¹(tt, V2) → ACTIVATE(V2)
U94¹(tt, V1, V2) → ISNAT(activate(V1))
U94¹(tt, V1, V2) → ACTIVATE(V1)
U94¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U93¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ACTIVATE(V1)
U92¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U92¹(tt, V1, V2) → ACTIVATE(V2)
U92¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ISNATKIND(activate(V1))
U91¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U114¹(tt, L) → ACTIVATE(L)
U113¹(tt, L, N) → ISNATKIND(activate(N))
U113¹(tt, L, N) → ACTIVATE(N)
U113¹(tt, L, N) → ACTIVATE(L)
U112¹(tt, L, N) → ISNAT(activate(N))
U112¹(tt, L, N) → ACTIVATE(N)
U112¹(tt, L, N) → ACTIVATE(L)
U111¹(tt, L, N) → ISNATILISTKIND(activate(L))
U111¹(tt, L, N) → ACTIVATE(L)
U111¹(tt, L, N) → ACTIVATE(N)
U51¹(tt, V2) → ACTIVATE(V2)
U121¹(tt, IL) → ACTIVATE(IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U131¹(isNatIList(activate(IL)), activate(IL), M, N)
U131¹(tt, IL, M, N) → U132¹(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132¹(tt, IL, M, N) → U133¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133¹(tt, IL, M, N) → U134¹(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134¹(tt, IL, M, N) → U135¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135¹(tt, IL, M, N) → U136¹(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136¹(tt, IL, M, N) → ACTIVATE(N)
U136¹(tt, IL, M, N) → ACTIVATE(M)
U136¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ISNATKIND(activate(N))
U135¹(tt, IL, M, N) → ACTIVATE(N)
U135¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ACTIVATE(M)
U134¹(tt, IL, M, N) → ISNAT(activate(N))
U134¹(tt, IL, M, N) → ACTIVATE(N)
U134¹(tt, IL, M, N) → ACTIVATE(IL)
U134¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ISNATKIND(activate(M))
U133¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ACTIVATE(IL)
U133¹(tt, IL, M, N) → ACTIVATE(N)
U132¹(tt, IL, M, N) → ISNAT(activate(M))
U132¹(tt, IL, M, N) → ACTIVATE(M)
U132¹(tt, IL, M, N) → ACTIVATE(IL)
U132¹(tt, IL, M, N) → ACTIVATE(N)
U131¹(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U131¹(tt, IL, M, N) → ACTIVATE(IL)
U131¹(tt, IL, M, N) → ACTIVATE(M)
U131¹(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
U11¹(tt, V1) → U12¹(isNatIListKind(activate(V1)), activate(V1))
U11¹(tt, V1) → ISNATILISTKIND(activate(V1))
U11¹(tt, V1) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x₁)) = x₁   
POL(ISNAT(x₁)) = x₁   
POL(ISNATILIST(x₁)) = x₁   
POL(ISNATILISTKIND(x₁)) = x₁   
POL(ISNATKIND(x₁)) = x₁   
POL(ISNATLIST(x₁)) = x₁   
POL(LENGTH(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = x₁ + x₂   
POL(U101(x₁, x₂, x₃)) = 0   
POL(U101¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U102(x₁, x₂, x₃)) = 0   
POL(U102¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U103(x₁, x₂, x₃)) = 0   
POL(U103¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U104(x₁, x₂, x₃)) = 0   
POL(U104¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U105(x₁, x₂)) = 0   
POL(U105¹(x₁, x₂)) = x₂   
POL(U106(x₁)) = 0   
POL(U11(x₁, x₂)) = 0   
POL(U111(x₁, x₂, x₃)) = 1 + x₂   
POL(U111¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U112(x₁, x₂, x₃)) = 1 + x₂   
POL(U112¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U113(x₁, x₂, x₃)) = 1 + x₂   
POL(U113¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U114(x₁, x₂)) = 1 + x₂   
POL(U114¹(x₁, x₂)) = x₂   
POL(U11¹(x₁, x₂)) = 1 + x₂   
POL(U12(x₁, x₂)) = 0   
POL(U121(x₁, x₂)) = 0   
POL(U121¹(x₁, x₂)) = x₂   
POL(U122(x₁)) = 0   
POL(U12¹(x₁, x₂)) = x₂   
POL(U13(x₁)) = 0   
POL(U131(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U131¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U132(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U132¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U133(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U133¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U134(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U134¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U135(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U135¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U136(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U136¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U21(x₁, x₂)) = 0   
POL(U21¹(x₁, x₂)) = x₂   
POL(U22(x₁, x₂)) = 0   
POL(U22¹(x₁, x₂)) = x₂   
POL(U23(x₁)) = 0   
POL(U31(x₁, x₂)) = 0   
POL(U31¹(x₁, x₂)) = x₂   
POL(U32(x₁, x₂)) = 0   
POL(U32¹(x₁, x₂)) = x₂   
POL(U33(x₁)) = 0   
POL(U41(x₁, x₂, x₃)) = 0   
POL(U41¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U42(x₁, x₂, x₃)) = 0   
POL(U42¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U43(x₁, x₂, x₃)) = 0   
POL(U43¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U44(x₁, x₂, x₃)) = 0   
POL(U44¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U45(x₁, x₂)) = 0   
POL(U45¹(x₁, x₂)) = x₂   
POL(U46(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 0   
POL(U61¹(x₁, x₂)) = x₂   
POL(U62(x₁)) = 0   
POL(U71(x₁)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃)) = 0   
POL(U91¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U92(x₁, x₂, x₃)) = 0   
POL(U92¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U93(x₁, x₂, x₃)) = 0   
POL(U93¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U94(x₁, x₂, x₃)) = 0   
POL(U94¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U95(x₁, x₂)) = 0   
POL(U95¹(x₁, x₂)) = x₂   
POL(U96(x₁)) = 0   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = 0   
POL(isNatIListKind(x₁)) = 0   
POL(isNatKind(x₁)) = 0   
POL(isNatList(x₁)) = 0   
POL(length(x₁)) = 1 + x₁   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 1 + x₁   
POL(n__nil) = 0   
POL(n__s(x₁)) = x₁   
POL(n__take(x₁, x₂)) = x₁ + x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = x₁   
POL(take(x₁, x₂)) = x₁ + x₂   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(6) Obligation:

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

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → U121¹(isNatIList(IL), IL)
U121¹(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
LENGTH(cons(N, L)) → U111¹(isNatList(activate(L)), activate(L), N)
U111¹(tt, L, N) → U112¹(isNatIListKind(activate(L)), activate(L), activate(N))
U112¹(tt, L, N) → U113¹(isNat(activate(N)), activate(L), activate(N))
U113¹(tt, L, N) → U114¹(isNatKind(activate(N)), activate(L))
U114¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U91¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U91¹(tt, V1, V2) → U92¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U92¹(tt, V1, V2) → U93¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93¹(tt, V1, V2) → U94¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94¹(tt, V1, V2) → U95¹(isNat(activate(V1)), activate(V2))
U95¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U101¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U101¹(tt, V1, V2) → U102¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U102¹(tt, V1, V2) → U103¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103¹(tt, V1, V2) → U104¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104¹(tt, V1, V2) → U105¹(isNat(activate(V1)), activate(V2))
U105¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U31¹(tt, V) → U32¹(isNatIListKind(activate(V)), activate(V))
U32¹(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U32¹(tt, V) → ACTIVATE(V)
U31¹(tt, V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, V) → ACTIVATE(V)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isNat(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U45¹(tt, V2) → ACTIVATE(V2)
U44¹(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
U12¹(tt, V1) → ISNATLIST(activate(V1))
U12¹(tt, V1) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U43¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ACTIVATE(V1)
U42¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U42¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U105¹(tt, V2) → ACTIVATE(V2)
U104¹(tt, V1, V2) → ISNAT(activate(V1))
U104¹(tt, V1, V2) → ACTIVATE(V1)
U104¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U103¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ACTIVATE(V1)
U102¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U102¹(tt, V1, V2) → ACTIVATE(V2)
U102¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ISNATKIND(activate(V1))
U101¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ACTIVATE(V2)
U95¹(tt, V2) → ACTIVATE(V2)
U94¹(tt, V1, V2) → ISNAT(activate(V1))
U94¹(tt, V1, V2) → ACTIVATE(V1)
U94¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U93¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ACTIVATE(V1)
U92¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U92¹(tt, V1, V2) → ACTIVATE(V2)
U92¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ISNATKIND(activate(V1))
U91¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U114¹(tt, L) → ACTIVATE(L)
U113¹(tt, L, N) → ISNATKIND(activate(N))
U113¹(tt, L, N) → ACTIVATE(N)
U113¹(tt, L, N) → ACTIVATE(L)
U112¹(tt, L, N) → ISNAT(activate(N))
U112¹(tt, L, N) → ACTIVATE(N)
U112¹(tt, L, N) → ACTIVATE(L)
U111¹(tt, L, N) → ISNATILISTKIND(activate(L))
U111¹(tt, L, N) → ACTIVATE(L)
U111¹(tt, L, N) → ACTIVATE(N)
U51¹(tt, V2) → ACTIVATE(V2)
U121¹(tt, IL) → ACTIVATE(IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U131¹(isNatIList(activate(IL)), activate(IL), M, N)
U131¹(tt, IL, M, N) → U132¹(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132¹(tt, IL, M, N) → U133¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133¹(tt, IL, M, N) → U134¹(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134¹(tt, IL, M, N) → U135¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135¹(tt, IL, M, N) → U136¹(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136¹(tt, IL, M, N) → ACTIVATE(N)
U136¹(tt, IL, M, N) → ACTIVATE(M)
U136¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ISNATKIND(activate(N))
U135¹(tt, IL, M, N) → ACTIVATE(N)
U135¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ACTIVATE(M)
U134¹(tt, IL, M, N) → ISNAT(activate(N))
U134¹(tt, IL, M, N) → ACTIVATE(N)
U134¹(tt, IL, M, N) → ACTIVATE(IL)
U134¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ISNATKIND(activate(M))
U133¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ACTIVATE(IL)
U133¹(tt, IL, M, N) → ACTIVATE(N)
U132¹(tt, IL, M, N) → ISNAT(activate(M))
U132¹(tt, IL, M, N) → ACTIVATE(M)
U132¹(tt, IL, M, N) → ACTIVATE(IL)
U132¹(tt, IL, M, N) → ACTIVATE(N)
U131¹(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U131¹(tt, IL, M, N) → ACTIVATE(IL)
U131¹(tt, IL, M, N) → ACTIVATE(M)
U131¹(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

TAKE(0, IL) → U121¹(isNatIList(IL), IL)
U121¹(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U31¹(tt, V) → U32¹(isNatIListKind(activate(V)), activate(V))
U32¹(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U91¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U91¹(tt, V1, V2) → U92¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U92¹(tt, V1, V2) → U93¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93¹(tt, V1, V2) → U94¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94¹(tt, V1, V2) → U95¹(isNat(activate(V1)), activate(V2))
U95¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U101¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U101¹(tt, V1, V2) → U102¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U102¹(tt, V1, V2) → U103¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103¹(tt, V1, V2) → U104¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104¹(tt, V1, V2) → U105¹(isNat(activate(V1)), activate(V2))
U105¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isNat(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U45¹(tt, V2) → ACTIVATE(V2)
U44¹(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U43¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ACTIVATE(V1)
U42¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U42¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U105¹(tt, V2) → ACTIVATE(V2)
U104¹(tt, V1, V2) → ISNAT(activate(V1))
U104¹(tt, V1, V2) → ACTIVATE(V1)
U104¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U103¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ACTIVATE(V1)
U102¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U102¹(tt, V1, V2) → ACTIVATE(V2)
U102¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ISNATKIND(activate(V1))
U101¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U95¹(tt, V2) → ACTIVATE(V2)
U94¹(tt, V1, V2) → ISNAT(activate(V1))
U94¹(tt, V1, V2) → ACTIVATE(V1)
U94¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U93¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ACTIVATE(V1)
U92¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U92¹(tt, V1, V2) → ACTIVATE(V2)
U92¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ISNATKIND(activate(V1))
U91¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ACTIVATE(V2)
U32¹(tt, V) → ACTIVATE(V)
U31¹(tt, V) → ISNATILISTKIND(activate(V))
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U131¹(isNatIList(activate(IL)), activate(IL), M, N)
U131¹(tt, IL, M, N) → U132¹(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132¹(tt, IL, M, N) → U133¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133¹(tt, IL, M, N) → U134¹(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134¹(tt, IL, M, N) → U135¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135¹(tt, IL, M, N) → U136¹(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136¹(tt, IL, M, N) → ACTIVATE(N)
U136¹(tt, IL, M, N) → ACTIVATE(M)
U136¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ISNATKIND(activate(N))
U135¹(tt, IL, M, N) → ACTIVATE(N)
U135¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ACTIVATE(M)
U134¹(tt, IL, M, N) → ISNAT(activate(N))
U134¹(tt, IL, M, N) → ACTIVATE(N)
U134¹(tt, IL, M, N) → ACTIVATE(IL)
U134¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ISNATKIND(activate(M))
U133¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ACTIVATE(IL)
U133¹(tt, IL, M, N) → ACTIVATE(N)
U132¹(tt, IL, M, N) → ISNAT(activate(M))
U132¹(tt, IL, M, N) → ACTIVATE(M)
U132¹(tt, IL, M, N) → ACTIVATE(IL)
U132¹(tt, IL, M, N) → ACTIVATE(N)
U131¹(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U131¹(tt, IL, M, N) → ACTIVATE(IL)
U131¹(tt, IL, M, N) → ACTIVATE(M)
U131¹(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U51¹(tt, V2) → ACTIVATE(V2)
U121¹(tt, IL) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(TAKE(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(0) =

/ 0 \ \ 0 /

POL(U1211(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(isNatIList(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(tt) =

/ 0 \ \ 0 /

POL(ISNATILISTKIND(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(n__cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U511(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(isNatKind(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(ISNATKIND(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(ACTIVATE(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(n__take(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(ISNATILIST(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(U311(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(isNatIListKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U321(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(ISNATLIST(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U911(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U921(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U931(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U941(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U951(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(isNat(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U1011(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U1021(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U1031(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U1041(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U1051(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U611(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U411(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U421(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U431(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U441(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(U451(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(ISNAT(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U211(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U221(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

POL(cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U1311(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U1321(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U1331(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U1341(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U1351(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U1361(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U101(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U102(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(zeros) =

/ 0 \ \ 0 /

POL(n__zeros) =

/ 0 \ \ 0 /

POL(U113(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U12(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U121(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U122(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(nil) =

/ 0 \ \ 0 /

POL(U131(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U103(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U11(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U111(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U43(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U33(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U135(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U31(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U23(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(n__0) =

/ 0 \ \ 0 /

POL(n__length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U61(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U51(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U52(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U71(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(take(x1, x2)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(n__nil) =

/ 0 \ \ 0 /

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatList(n__nil) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__0) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(11) Obligation:

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

TAKE(0, IL) → U121¹(isNatIList(IL), IL)
U121¹(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U31¹(tt, V) → U32¹(isNatIListKind(activate(V)), activate(V))
U32¹(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U91¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U91¹(tt, V1, V2) → U92¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U92¹(tt, V1, V2) → U93¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93¹(tt, V1, V2) → U94¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94¹(tt, V1, V2) → U95¹(isNat(activate(V1)), activate(V2))
U95¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U101¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U101¹(tt, V1, V2) → U102¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U102¹(tt, V1, V2) → U103¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103¹(tt, V1, V2) → U104¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104¹(tt, V1, V2) → U105¹(isNat(activate(V1)), activate(V2))
U105¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))
U61¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isNat(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U45¹(tt, V2) → ACTIVATE(V2)
U44¹(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U22¹(tt, V1) → ACTIVATE(V1)
U21¹(tt, V1) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V1)
U44¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U43¹(tt, V1, V2) → ACTIVATE(V2)
U43¹(tt, V1, V2) → ACTIVATE(V1)
U42¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U42¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U105¹(tt, V2) → ACTIVATE(V2)
U104¹(tt, V1, V2) → ISNAT(activate(V1))
U104¹(tt, V1, V2) → ACTIVATE(V1)
U104¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U103¹(tt, V1, V2) → ACTIVATE(V2)
U103¹(tt, V1, V2) → ACTIVATE(V1)
U102¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U102¹(tt, V1, V2) → ACTIVATE(V2)
U102¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ISNATKIND(activate(V1))
U101¹(tt, V1, V2) → ACTIVATE(V1)
U101¹(tt, V1, V2) → ACTIVATE(V2)
U95¹(tt, V2) → ACTIVATE(V2)
U94¹(tt, V1, V2) → ISNAT(activate(V1))
U94¹(tt, V1, V2) → ACTIVATE(V1)
U94¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U93¹(tt, V1, V2) → ACTIVATE(V2)
U93¹(tt, V1, V2) → ACTIVATE(V1)
U92¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U92¹(tt, V1, V2) → ACTIVATE(V2)
U92¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ISNATKIND(activate(V1))
U91¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, V1, V2) → ACTIVATE(V2)
U32¹(tt, V) → ACTIVATE(V)
U31¹(tt, V) → ISNATILISTKIND(activate(V))
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U131¹(isNatIList(activate(IL)), activate(IL), M, N)
U131¹(tt, IL, M, N) → U132¹(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132¹(tt, IL, M, N) → U133¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133¹(tt, IL, M, N) → U134¹(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134¹(tt, IL, M, N) → U135¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135¹(tt, IL, M, N) → U136¹(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136¹(tt, IL, M, N) → ACTIVATE(N)
U136¹(tt, IL, M, N) → ACTIVATE(M)
U136¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ISNATKIND(activate(N))
U135¹(tt, IL, M, N) → ACTIVATE(N)
U135¹(tt, IL, M, N) → ACTIVATE(IL)
U135¹(tt, IL, M, N) → ACTIVATE(M)
U134¹(tt, IL, M, N) → ISNAT(activate(N))
U134¹(tt, IL, M, N) → ACTIVATE(N)
U134¹(tt, IL, M, N) → ACTIVATE(IL)
U134¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ISNATKIND(activate(M))
U133¹(tt, IL, M, N) → ACTIVATE(M)
U133¹(tt, IL, M, N) → ACTIVATE(IL)
U133¹(tt, IL, M, N) → ACTIVATE(N)
U132¹(tt, IL, M, N) → ISNAT(activate(M))
U132¹(tt, IL, M, N) → ACTIVATE(M)
U132¹(tt, IL, M, N) → ACTIVATE(IL)
U132¹(tt, IL, M, N) → ACTIVATE(N)
U131¹(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U131¹(tt, IL, M, N) → ACTIVATE(IL)
U131¹(tt, IL, M, N) → ACTIVATE(M)
U131¹(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U51¹(tt, V2) → ACTIVATE(V2)
U121¹(tt, IL) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 96 less nodes.

(13) Complex Obligation (AND)

(14) Obligation:

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(15) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

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

(17) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
  The graph contains the following edges 1 > 1

• ACTIVATE(n__s(X)) → ACTIVATE(X)
  The graph contains the following edges 1 > 1

(18) TRUE

(19) Obligation:

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

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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.

ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(ISNATKIND(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(U101(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(tt) =

/ 1 \ \ 0 /

POL(U102(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(isNatKind(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(zeros) =

/ 1 \ \ 0 /

POL(cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(0) =

/ 0 \ \ 0 /

POL(n__zeros) =

/ 1 \ \ 0 /

POL(U113(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(s(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U12(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U121(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U122(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIListKind(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(nil) =

/ 0 \ \ 1 /

POL(U131(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(isNat(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U103(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 1 0 /

·

x1

POL(U11(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U111(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 1 1 /

·

x3

POL(U43(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 1 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 1 1 /

·

x3

POL(U33(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 1 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(n__take(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U135(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U31(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 0 /

·

x2

POL(U23(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(n__0) =

/ 0 \ \ 0 /

POL(n__length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U61(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U51(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U52(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(U71(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(take(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(n__nil) =

/ 0 \ \ 1 /

POL(n__cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatList(n__nil) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(21) Obligation:

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

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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:

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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.

ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(ISNAT(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U211(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(isNatKind(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(tt) =

/ 0 \ \ 1 /

POL(U221(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(U101(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U102(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(zeros) =

/ 0 \ \ 1 /

POL(cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 1 /

·

x2

POL(0) =

/ 0 \ \ 0 /

POL(n__zeros) =

/ 0 \ \ 1 /

POL(U113(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(s(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U12(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U121(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U122(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(isNatIListKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(nil) =

/ 1 \ \ 1 /

POL(U131(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(isNat(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U103(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U11(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U111(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

POL(U43(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 1 /

·

x3

POL(U33(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(n__take(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U135(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U31(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U23(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(n__0) =

/ 0 \ \ 0 /

POL(n__length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U61(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U62(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U51(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U52(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U71(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(take(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(n__nil) =

/ 1 \ \ 1 /

POL(n__cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 1 /

·

x2

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatList(n__nil) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__0) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__zeros) → tt
isNatIListKind(n__nil) → tt
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(26) Obligation:

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

U21¹(tt, V1) → U22¹(isNatKind(activate(V1)), activate(V1))
U22¹(tt, V1) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(28) TRUE

(29) Obligation:

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

U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__take(V1, V2)) → U61¹(isNatKind(activate(V1)), activate(V2))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ISNATILISTKIND(x₁)) = x₁   
POL(U101(x₁, x₂, x₃)) = 1   
POL(U102(x₁, x₂, x₃)) = 1   
POL(U103(x₁, x₂, x₃)) = 1   
POL(U104(x₁, x₂, x₃)) = 1   
POL(U105(x₁, x₂)) = 0   
POL(U106(x₁)) = 0   
POL(U11(x₁, x₂)) = 1   
POL(U111(x₁, x₂, x₃)) = 0   
POL(U112(x₁, x₂, x₃)) = 0   
POL(U113(x₁, x₂, x₃)) = 0   
POL(U114(x₁, x₂)) = 0   
POL(U12(x₁, x₂)) = 1   
POL(U121(x₁, x₂)) = 0   
POL(U122(x₁)) = 0   
POL(U13(x₁)) = 1   
POL(U131(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U132(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U133(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U134(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U135(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U136(x₁, x₂, x₃, x₄)) = 1 + x₂   
POL(U21(x₁, x₂)) = 0   
POL(U22(x₁, x₂)) = 0   
POL(U23(x₁)) = 0   
POL(U31(x₁, x₂)) = 0   
POL(U32(x₁, x₂)) = 0   
POL(U33(x₁)) = 0   
POL(U41(x₁, x₂, x₃)) = 0   
POL(U42(x₁, x₂, x₃)) = 0   
POL(U43(x₁, x₂, x₃)) = 0   
POL(U44(x₁, x₂, x₃)) = 0   
POL(U45(x₁, x₂)) = 0   
POL(U46(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 0   
POL(U61¹(x₁, x₂)) = x₂   
POL(U62(x₁)) = 0   
POL(U71(x₁)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃)) = 0   
POL(U92(x₁, x₂, x₃)) = 0   
POL(U93(x₁, x₂, x₃)) = 0   
POL(U94(x₁, x₂, x₃)) = 0   
POL(U95(x₁, x₂)) = 0   
POL(U96(x₁)) = 0   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₂   
POL(isNat(x₁)) = 1   
POL(isNatIList(x₁)) = x₁   
POL(isNatIListKind(x₁)) = 0   
POL(isNatKind(x₁)) = 0   
POL(isNatList(x₁)) = 1   
POL(length(x₁)) = 1   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₂   
POL(n__length(x₁)) = 1   
POL(n__nil) = 0   
POL(n__s(x₁)) = 0   
POL(n__take(x₁, x₂)) = 1 + x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = 1 + x₂   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(31) Obligation:

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

U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILISTKIND(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(32) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(33) Obligation:

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

ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(34) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))

(35) Obligation:

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

U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(36) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, V2) → ISNATILISTKIND(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
U51¹(tt, x0) → ISNATILISTKIND(x0)

(37) Obligation:

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

ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
U51¹(tt, x0) → ISNATILISTKIND(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(38) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__zeros), activate(y0))

(39) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__zeros), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(40) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(41) Obligation:

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

U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(42) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__zeros) → ISNATILISTKIND(zeros) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__zeros)

(43) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(44) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(45) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(46) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__0) → ISNATILISTKIND(0) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__0) → ISNATILISTKIND(n__0)

(47) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__0) → ISNATILISTKIND(n__0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(49) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(50) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))

(51) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(52) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__nil) → ISNATILISTKIND(nil) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__nil) → ISNATILISTKIND(n__nil)

(53) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__nil) → ISNATILISTKIND(n__nil)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(54) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(55) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(56) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILISTKIND(n__cons(n__nil, y0)) → U51¹(isNatKind(n__nil), activate(y0))

(57) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
ISNATILISTKIND(n__cons(n__nil, y0)) → U51¹(isNatKind(n__nil), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(58) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(59) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(60) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(n__0, n__zeros)), activate(y0))

(61) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(62) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(63) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(64) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))

(65) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(66) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__cons(n__0, n__zeros)), activate(y0))

(67) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(68) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(69) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(70) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

(71) Obligation:

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

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__take(x0, x1), y1)) → U51¹(isNatKind(take(activate(x0), activate(x1))), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ISNATILISTKIND(x₁)) = x₁   
POL(U101(x₁, x₂, x₃)) = 1   
POL(U102(x₁, x₂, x₃)) = 1   
POL(U103(x₁, x₂, x₃)) = 1   
POL(U104(x₁, x₂, x₃)) = 1   
POL(U105(x₁, x₂)) = 0   
POL(U106(x₁)) = 0   
POL(U11(x₁, x₂)) = 0   
POL(U111(x₁, x₂, x₃)) = 0   
POL(U112(x₁, x₂, x₃)) = 0   
POL(U113(x₁, x₂, x₃)) = 0   
POL(U114(x₁, x₂)) = 0   
POL(U12(x₁, x₂)) = 0   
POL(U121(x₁, x₂)) = 0   
POL(U122(x₁)) = 0   
POL(U13(x₁)) = 0   
POL(U131(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U132(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U133(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U134(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U135(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U136(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U21(x₁, x₂)) = 0   
POL(U22(x₁, x₂)) = 0   
POL(U23(x₁)) = 0   
POL(U31(x₁, x₂)) = 0   
POL(U32(x₁, x₂)) = 0   
POL(U33(x₁)) = 0   
POL(U41(x₁, x₂, x₃)) = x₂   
POL(U42(x₁, x₂, x₃)) = x₂   
POL(U43(x₁, x₂, x₃)) = x₂   
POL(U44(x₁, x₂, x₃)) = 0   
POL(U45(x₁, x₂)) = 0   
POL(U46(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 0   
POL(U62(x₁)) = 0   
POL(U71(x₁)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃)) = 1   
POL(U92(x₁, x₂, x₃)) = 1   
POL(U93(x₁, x₂, x₃)) = 0   
POL(U94(x₁, x₂, x₃)) = 0   
POL(U95(x₁, x₂)) = 0   
POL(U96(x₁)) = 0   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = x₁   
POL(isNatIListKind(x₁)) = 0   
POL(isNatKind(x₁)) = 0   
POL(isNatList(x₁)) = 1   
POL(length(x₁)) = 0   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 0   
POL(n__nil) = 0   
POL(n__s(x₁)) = 0   
POL(n__take(x₁, x₂)) = 1 + x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = 1 + x₂   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(73) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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.

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(activate(x0))), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ISNATILISTKIND(x₁)) = x₁   
POL(U101(x₁, x₂, x₃)) = x₃   
POL(U102(x₁, x₂, x₃)) = 0   
POL(U103(x₁, x₂, x₃)) = 0   
POL(U104(x₁, x₂, x₃)) = 0   
POL(U105(x₁, x₂)) = 0   
POL(U106(x₁)) = 0   
POL(U11(x₁, x₂)) = 0   
POL(U111(x₁, x₂, x₃)) = 0   
POL(U112(x₁, x₂, x₃)) = 0   
POL(U113(x₁, x₂, x₃)) = 0   
POL(U114(x₁, x₂)) = 0   
POL(U12(x₁, x₂)) = 0   
POL(U121(x₁, x₂)) = 0   
POL(U122(x₁)) = 0   
POL(U13(x₁)) = 0   
POL(U131(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U132(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U133(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U134(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U135(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U136(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U21(x₁, x₂)) = 0   
POL(U22(x₁, x₂)) = 0   
POL(U23(x₁)) = 0   
POL(U31(x₁, x₂)) = 1   
POL(U32(x₁, x₂)) = 1   
POL(U33(x₁)) = 0   
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃   
POL(U42(x₁, x₂, x₃)) = 1   
POL(U43(x₁, x₂, x₃)) = 1   
POL(U44(x₁, x₂, x₃)) = 1   
POL(U45(x₁, x₂)) = x₁   
POL(U46(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 0   
POL(U62(x₁)) = 0   
POL(U71(x₁)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃)) = 0   
POL(U92(x₁, x₂, x₃)) = 0   
POL(U93(x₁, x₂, x₃)) = 0   
POL(U94(x₁, x₂, x₃)) = 0   
POL(U95(x₁, x₂)) = 0   
POL(U96(x₁)) = 0   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 1   
POL(isNatIList(x₁)) = 1 + x₁   
POL(isNatIListKind(x₁)) = 1 + x₁   
POL(isNatKind(x₁)) = 0   
POL(isNatList(x₁)) = 1 + x₁   
POL(length(x₁)) = 1 + x₁   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 1 + x₁   
POL(n__nil) = 0   
POL(n__s(x₁)) = 0   
POL(n__take(x₁, x₂)) = x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = x₂   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(75) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → 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.

ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(activate(x0))), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ISNATILISTKIND(x₁)) = x₁   
POL(U101(x₁, x₂, x₃)) = 0   
POL(U102(x₁, x₂, x₃)) = 0   
POL(U103(x₁, x₂, x₃)) = 0   
POL(U104(x₁, x₂, x₃)) = 0   
POL(U105(x₁, x₂)) = 0   
POL(U106(x₁)) = 0   
POL(U11(x₁, x₂)) = 0   
POL(U111(x₁, x₂, x₃)) = 1   
POL(U112(x₁, x₂, x₃)) = 1   
POL(U113(x₁, x₂, x₃)) = 1   
POL(U114(x₁, x₂)) = 1   
POL(U12(x₁, x₂)) = 0   
POL(U121(x₁, x₂)) = 0   
POL(U122(x₁)) = 0   
POL(U13(x₁)) = 0   
POL(U131(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U132(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U133(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U134(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U135(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U136(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U21(x₁, x₂)) = 0   
POL(U22(x₁, x₂)) = 0   
POL(U23(x₁)) = 0   
POL(U31(x₁, x₂)) = 0   
POL(U32(x₁, x₂)) = 0   
POL(U33(x₁)) = 0   
POL(U41(x₁, x₂, x₃)) = 0   
POL(U42(x₁, x₂, x₃)) = 0   
POL(U43(x₁, x₂, x₃)) = 0   
POL(U44(x₁, x₂, x₃)) = 0   
POL(U45(x₁, x₂)) = 0   
POL(U46(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 0   
POL(U62(x₁)) = 0   
POL(U71(x₁)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃)) = 0   
POL(U92(x₁, x₂, x₃)) = 0   
POL(U93(x₁, x₂, x₃)) = 0   
POL(U94(x₁, x₂, x₃)) = 0   
POL(U95(x₁, x₂)) = 0   
POL(U96(x₁)) = 0   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = 0   
POL(isNatIListKind(x₁)) = x₁   
POL(isNatKind(x₁)) = 0   
POL(isNatList(x₁)) = 0   
POL(length(x₁)) = 1   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 1   
POL(n__nil) = 0   
POL(n__s(x₁)) = 1   
POL(n__take(x₁, x₂)) = x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 1   
POL(take(x₁, x₂)) = x₂   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(77) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U511(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 0 /

·

x2

POL(tt) =

/ 0 \ \ 0 /

POL(n__take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(ISNATILISTKIND(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(n__length(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(length(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(n__cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(isNatKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(n__0) =

/ 0 \ \ 1 /

POL(n__zeros) =

/ 0 \ \ 0 /

POL(0) =

/ 0 \ \ 1 /

POL(U101(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U102(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(zeros) =

/ 0 \ \ 0 /

POL(U113(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U12(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U121(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U122(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIListKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(nil) =

/ 0 \ \ 0 /

POL(U131(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(isNat(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U103(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U11(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U111(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 1 /

·

x3

POL(U43(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 1 /

·

x3

POL(U33(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U135(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U31(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U23(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

POL(U61(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U51(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U52(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

POL(U71(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__nil) =

/ 0 \ \ 0 /

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(79) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(activate(x0), x1)), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U511(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(tt) =

/ 0 \ \ 0 /

POL(n__take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(ISNATILISTKIND(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__cons(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(isNatKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(cons(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(n__0) =

/ 0 \ \ 0 /

POL(n__zeros) =

/ 1 \ \ 0 /

POL(0) =

/ 0 \ \ 0 /

POL(U101(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U102(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(zeros) =

/ 1 \ \ 0 /

POL(U113(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(length(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U12(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U121(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U122(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIListKind(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(nil) =

/ 0 \ \ 0 /

POL(U131(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 1 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 1 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 1 0 /

·

x4

POL(isNat(x1)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U103(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U11(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U111(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(U43(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

POL(U33(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 1 0 /

·

x4

POL(U135(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 1 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 1 0 /

·

x4

POL(U31(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U23(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(n__length(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U61(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U51(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U52(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U71(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__nil) =

/ 0 \ \ 0 /

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIList(n__zeros) → tt
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(81) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U511(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(tt) =

/ 0 \ \ 0 /

POL(n__take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(ISNATILISTKIND(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(n__s(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(s(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

POL(isNatKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__0) =

/ 0 \ \ 0 /

POL(n__zeros) =

/ 0 \ \ 0 /

POL(0) =

/ 0 \ \ 0 /

POL(U101(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U102(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(zeros) =

/ 0 \ \ 0 /

POL(U113(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(length(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U12(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U121(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U122(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIListKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(nil) =

/ 0 \ \ 0 /

POL(U131(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(isNat(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U103(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 1 /

·

x1

POL(U11(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U111(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 0 /

·

x3

POL(U43(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 1 /

·

x3

POL(U33(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U135(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U31(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U23(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(n__length(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(U61(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U51(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U52(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U71(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__nil) =

/ 0 \ \ 0 /

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(83) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__take(x0, x1)) → ISNATILISTKIND(take(activate(x0), activate(x1)))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U511(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

POL(tt) =

/ 1 \ \ 0 /

POL(n__take(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(ISNATILISTKIND(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(take(x1, x2)) =

/ 0 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(n__cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(cons(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

POL(isNatKind(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(n__0) =

/ 1 \ \ 0 /

POL(n__zeros) =

/ 0 \ \ 0 /

POL(0) =

/ 1 \ \ 0 /

POL(U101(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U102(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(zeros) =

/ 0 \ \ 0 /

POL(U113(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U114(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(s(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(length(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U12(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U13(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(U121(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U122(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIListKind(x1)) =

/ 1 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(nil) =

/ 1 \ \ 1 /

POL(U131(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U132(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U133(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(isNat(x1)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(U103(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U104(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U105(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U106(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 1 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U11(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U111(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U112(x1, x2, x3)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U42(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 1 /

·

x3

POL(U43(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 0 /

·

x3

POL(U41(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 1 /

·

x3

POL(U33(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U32(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U46(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U45(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U44(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U136(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U135(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U134(x1, x2, x3, x4)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(U31(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U23(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U22(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U21(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(U95(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U96(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U93(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 1 /

·

x3

POL(U94(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 1 /

·

x3

POL(n__s(x1)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 0 /

·

x1

POL(n__length(x1)) =

/ 0 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U61(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U51(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

POL(U52(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 1 /

·

x3

POL(U92(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 1 /

·

x3

POL(U71(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U81(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__nil) =

/ 1 \ \ 1 /

The following usable rules [FROCOS05] were oriented:

U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
zeros → cons(0, n__zeros)
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U33(tt) → tt
U32(tt, V) → U33(isNatList(activate(V)))
U46(tt) → tt
U45(tt, V2) → U46(isNatIList(activate(V2)))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U23(tt) → tt
U22(tt, V1) → U23(isNat(activate(V1)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U71(tt) → tt
U81(tt) → tt
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
length(nil) → 0
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
take(0, IL) → U121(isNatIList(IL), IL)
isNatList(n__nil) → tt
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)

(85) Obligation:

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

U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(86) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U51¹(isNatKind(n__0), activate(n__zeros)) evaluates to t =U51¹(isNatKind(n__0), activate(n__zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U51¹(isNatKind(n__0), activate(n__zeros)) → U51¹(isNatKind(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U51¹(isNatKind(n__0), n__zeros) → U51¹(tt, n__zeros)
with rule isNatKind(n__0) → tt at position [0] and matcher [ ]

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
with rule U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATILISTKIND(n__cons(n__0, n__zeros)) → U51¹(isNatKind(n__0), activate(n__zeros))
with rule ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(87) FALSE

(88) Obligation:

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

ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U31¹(tt, V) → U32¹(isNatIListKind(activate(V)), activate(V))
U32¹(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U91¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U91¹(tt, V1, V2) → U92¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U92¹(tt, V1, V2) → U93¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93¹(tt, V1, V2) → U94¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94¹(tt, V1, V2) → U95¹(isNat(activate(V1)), activate(V2))
U95¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → U101¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U101¹(tt, V1, V2) → U102¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U102¹(tt, V1, V2) → U103¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103¹(tt, V1, V2) → U104¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104¹(tt, V1, V2) → U105¹(isNat(activate(V1)), activate(V2))
U105¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43¹(tt, V1, V2) → U44¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44¹(tt, V1, V2) → U45¹(isNat(activate(V1)), activate(V2))
U45¹(tt, V2) → ISNATILIST(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(89) Obligation:

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

LENGTH(cons(N, L)) → U111¹(isNatList(activate(L)), activate(L), N)
U111¹(tt, L, N) → U112¹(isNatIListKind(activate(L)), activate(L), activate(N))
U112¹(tt, L, N) → U113¹(isNat(activate(N)), activate(L), activate(N))
U113¹(tt, L, N) → U114¹(isNatKind(activate(N)), activate(L))
U114¹(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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