(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U11(tt, V1)) → U121(isNatList(V1))
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(U21(tt, V1)) → ISNAT(V1)
ACTIVE(U31(tt, V)) → U321(isNatList(V))
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ACTIVE(U41(tt, V1, V2)) → U421(isNat(V1), V2)
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U42(tt, V2)) → U431(isNatIList(V2))
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
ACTIVE(U51(tt, V1, V2)) → U521(isNat(V1), V2)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U52(tt, V2)) → U531(isNatList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
ACTIVE(U61(tt, V1, V2)) → U621(isNat(V1), V2)
ACTIVE(U61(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U62(tt, V2)) → U631(isNatIList(V2))
ACTIVE(U62(tt, V2)) → ISNATILIST(V2)
ACTIVE(U71(tt, L)) → S(length(L))
ACTIVE(U71(tt, L)) → LENGTH(L)
ACTIVE(U91(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(U91(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(isNat(length(V1))) → U111(isNatIListKind(V1), V1)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatIList(V)) → U311(isNatIListKind(V), V)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNatIList(cons(V1, V2))) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(isNatIListKind(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatList(cons(V1, V2))) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(isNatList(take(V1, V2))) → U611(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(isNatList(take(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(isNatList(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatList(take(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(length(cons(N, L))) → U711(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
ACTIVE(take(0, IL)) → U811(and(isNatIList(IL), isNatIListKind(IL)))
ACTIVE(take(0, IL)) → AND(isNatIList(IL), isNatIListKind(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(take(0, IL)) → ISNATILISTKIND(IL)
ACTIVE(take(s(M), cons(N, IL))) → U911(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), isNatIListKind(IL))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILISTKIND(IL)
ACTIVE(take(s(M), cons(N, IL))) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNatKind(M))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(M)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatKind(N))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
ACTIVE(take(s(M), cons(N, IL))) → ISNATKIND(N)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → U211(active(X1), X2)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → U221(active(X))
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → U321(active(X))
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → U411(active(X1), X2, X3)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U42(X1, X2)) → U421(active(X1), X2)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → U431(active(X))
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → U521(active(X1), X2)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → U531(active(X))
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → U611(active(X1), X2, X3)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → U621(active(X1), X2)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → U631(active(X))
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → U711(active(X1), X2)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U81(X)) → U811(active(X))
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2, X3, X4)) → U911(active(X1), X2, X3, X4)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
CONS(mark(X1), X2) → CONS(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U121(mark(X)) → U121(X)
U211(mark(X1), X2) → U211(X1, X2)
U221(mark(X)) → U221(X)
U311(mark(X1), X2) → U311(X1, X2)
U321(mark(X)) → U321(X)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
U421(mark(X1), X2) → U421(X1, X2)
U431(mark(X)) → U431(X)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U521(mark(X1), X2) → U521(X1, X2)
U531(mark(X)) → U531(X)
U611(mark(X1), X2, X3) → U611(X1, X2, X3)
U621(mark(X1), X2) → U621(X1, X2)
U631(mark(X)) → U631(X)
U711(mark(X1), X2) → U711(X1, X2)
S(mark(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
U811(mark(X)) → U811(X)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X)) → U121(proper(X))
PROPER(U12(X)) → PROPER(X)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(isNatList(X)) → PROPER(X)
PROPER(U21(X1, X2)) → U211(proper(X1), proper(X2))
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → U221(proper(X))
PROPER(U22(X)) → PROPER(X)
PROPER(isNat(X)) → ISNAT(proper(X))
PROPER(isNat(X)) → PROPER(X)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X)) → U321(proper(X))
PROPER(U32(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → U411(proper(X1), proper(X2), proper(X3))
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2)) → U421(proper(X1), proper(X2))
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → U431(proper(X))
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2)) → U521(proper(X1), proper(X2))
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U53(X)) → U531(proper(X))
PROPER(U53(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → U611(proper(X1), proper(X2), proper(X3))
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2)) → U621(proper(X1), proper(X2))
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → U631(proper(X))
PROPER(U63(X)) → PROPER(X)
PROPER(U71(X1, X2)) → U711(proper(X1), proper(X2))
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → U811(proper(X))
PROPER(U81(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → U911(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatIListKind(X)) → ISNATILISTKIND(proper(X))
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(isNatKind(X)) → ISNATKIND(proper(X))
PROPER(isNatKind(X)) → PROPER(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
U111(ok(X1), ok(X2)) → U111(X1, X2)
U121(ok(X)) → U121(X)
ISNATLIST(ok(X)) → ISNATLIST(X)
U211(ok(X1), ok(X2)) → U211(X1, X2)
U221(ok(X)) → U221(X)
ISNAT(ok(X)) → ISNAT(X)
U311(ok(X1), ok(X2)) → U311(X1, X2)
U321(ok(X)) → U321(X)
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U421(ok(X1), ok(X2)) → U421(X1, X2)
U431(ok(X)) → U431(X)
ISNATILIST(ok(X)) → ISNATILIST(X)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U521(ok(X1), ok(X2)) → U521(X1, X2)
U531(ok(X)) → U531(X)
U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
U621(ok(X1), ok(X2)) → U621(X1, X2)
U631(ok(X)) → U631(X)
U711(ok(X1), ok(X2)) → U711(X1, X2)
S(ok(X)) → S(X)
LENGTH(ok(X)) → LENGTH(X)
U811(ok(X)) → U811(X)
U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
ISNATKIND(ok(X)) → ISNATKIND(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 31 SCCs with 127 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNATKIND(ok(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(ok(X)) → ISNATKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNATKIND1

Status:
ISNATKIND1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNATILISTKIND1

Status:
ISNATILISTKIND1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

ISNATILIST(ok(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATILIST(ok(X)) → ISNATILIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNATILIST1

Status:
ISNATILIST1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(17) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNAT(ok(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNAT1

Status:
ok1: multiset
ISNAT1: multiset


The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) Obligation:

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

ISNATLIST(ok(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISNATLIST(ok(X)) → ISNATLIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > ISNATLIST1

Status:
ISNATLIST1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(ok(X1), ok(X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > AND1
mark > AND1

Status:
AND1: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

AND(mark(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AND(mark(X1), X2) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[AND1, mark1]

Status:
AND1: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x1, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
TAKE2: [2,1]
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[TAKE1, ok1]

Status:
TAKE1: multiset
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(ok(X1), ok(X2), ok(X3), ok(X4)) → U911(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U911(x1, x2, x3, x4)  =  U911(x3, x4)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U91^12: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(mark(X1), X2, X3, X4) → U911(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U91^14: [2,1,3,4]
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

U811(ok(X)) → U811(X)
U811(mark(X)) → U811(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U811(ok(X)) → U811(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U811(x1)  =  U811(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U81^11, ok1]

Status:
U81^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U811(mark(X)) → U811(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U811(mark(X)) → U811(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U81^11

Status:
U81^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) Obligation:

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

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LENGTH(ok(X)) → LENGTH(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  LENGTH(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[LENGTH1, ok1]

Status:
LENGTH1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

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

LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LENGTH(mark(X)) → LENGTH(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > LENGTH1

Status:
LENGTH1: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(63) PisEmptyProof (EQUIVALENT transformation)

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

(64) TRUE

(65) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(ok(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[S1, ok1]

Status:
ok1: multiset
S1: multiset


The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > S1

Status:
mark1: multiset
S1: multiset


The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(ok(X1), ok(X2)) → U711(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2)  =  U711(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U71^11
mark > U71^11

Status:
U71^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

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

U711(mark(X1), X2) → U711(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(mark(X1), X2) → U711(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2)  =  U711(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U71^11, mark1]

Status:
U71^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(76) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(77) PisEmptyProof (EQUIVALENT transformation)

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

(78) TRUE

(79) Obligation:

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

U631(ok(X)) → U631(X)
U631(mark(X)) → U631(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U631(ok(X)) → U631(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U631(x1)  =  U631(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U63^11, ok1]

Status:
U63^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

U631(mark(X)) → U631(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U631(mark(X)) → U631(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U63^11

Status:
U63^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(84) PisEmptyProof (EQUIVALENT transformation)

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

(85) TRUE

(86) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(ok(X1), ok(X2)) → U621(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2)  =  U621(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U62^11
mark > U62^11

Status:
mark: multiset
ok1: multiset
U62^11: multiset


The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

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

U621(mark(X1), X2) → U621(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(89) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(mark(X1), X2) → U621(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2)  =  U621(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U62^11, mark1]

Status:
mark1: [1]
U62^11: multiset


The following usable rules [FROCOS05] were oriented: none

(90) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(91) PisEmptyProof (EQUIVALENT transformation)

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

(92) TRUE

(93) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2, x3)  =  U611(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U61^12, ok1]

Status:
U61^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X1), X2, X3) → U611(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2, x3)  =  U611(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U61^11, mark1]

Status:
U61^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(97) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(98) PisEmptyProof (EQUIVALENT transformation)

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

(99) TRUE

(100) Obligation:

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

U531(ok(X)) → U531(X)
U531(mark(X)) → U531(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U531(ok(X)) → U531(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U531(x1)  =  U531(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U53^11, ok1]

Status:
ok1: multiset
U53^11: multiset


The following usable rules [FROCOS05] were oriented: none

(102) Obligation:

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

U531(mark(X)) → U531(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(103) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U531(mark(X)) → U531(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U53^11

Status:
mark1: multiset
U53^11: multiset


The following usable rules [FROCOS05] were oriented: none

(104) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(105) PisEmptyProof (EQUIVALENT transformation)

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

(106) TRUE

(107) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(ok(X1), ok(X2)) → U521(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U521(x1, x2)  =  U521(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U52^11
mark > U52^11

Status:
U52^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

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

U521(mark(X1), X2) → U521(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(110) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(mark(X1), X2) → U521(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U521(x1, x2)  =  U521(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U52^11, mark1]

Status:
U52^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(111) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(112) PisEmptyProof (EQUIVALENT transformation)

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

(113) TRUE

(114) Obligation:

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

U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U51^12, ok1]

Status:
U51^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

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

U511(mark(X1), X2, X3) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(mark(X1), X2, X3) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2, x3)  =  U511(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U51^11, mark1]

Status:
mark1: multiset
U51^11: multiset


The following usable rules [FROCOS05] were oriented: none

(118) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(119) PisEmptyProof (EQUIVALENT transformation)

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

(120) TRUE

(121) Obligation:

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

U431(ok(X)) → U431(X)
U431(mark(X)) → U431(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U431(ok(X)) → U431(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U431(x1)  =  U431(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U43^11, ok1]

Status:
ok1: multiset
U43^11: multiset


The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

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

U431(mark(X)) → U431(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U431(mark(X)) → U431(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U43^11

Status:
mark1: multiset
U43^11: multiset


The following usable rules [FROCOS05] were oriented: none

(125) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(126) PisEmptyProof (EQUIVALENT transformation)

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

(127) TRUE

(128) Obligation:

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

U421(ok(X1), ok(X2)) → U421(X1, X2)
U421(mark(X1), X2) → U421(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U421(ok(X1), ok(X2)) → U421(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U421(x1, x2)  =  U421(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U42^11
mark > U42^11

Status:
mark: multiset
ok1: multiset
U42^11: multiset


The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

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

U421(mark(X1), X2) → U421(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U421(mark(X1), X2) → U421(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U421(x1, x2)  =  U421(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U42^11, mark1]

Status:
mark1: [1]
U42^11: multiset


The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(133) PisEmptyProof (EQUIVALENT transformation)

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

(134) TRUE

(135) Obligation:

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

U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2, x3)  =  U411(x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U41^12, ok1]

Status:
U41^12: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(137) Obligation:

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

U411(mark(X1), X2, X3) → U411(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(138) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(mark(X1), X2, X3) → U411(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2, x3)  =  U411(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U41^11, mark1]

Status:
U41^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(139) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(140) PisEmptyProof (EQUIVALENT transformation)

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

(141) TRUE

(142) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(ok(X)) → U321(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U321(x1)  =  U321(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U32^11, ok1]

Status:
ok1: multiset
U32^11: multiset


The following usable rules [FROCOS05] were oriented: none

(144) Obligation:

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

U321(mark(X)) → U321(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(145) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(mark(X)) → U321(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U32^11

Status:
mark1: multiset
U32^11: multiset


The following usable rules [FROCOS05] were oriented: none

(146) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(147) PisEmptyProof (EQUIVALENT transformation)

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

(148) TRUE

(149) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(150) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(ok(X1), ok(X2)) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2)  =  U311(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U31^11
mark > U31^11

Status:
U31^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(151) Obligation:

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

U311(mark(X1), X2) → U311(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(152) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(mark(X1), X2) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2)  =  U311(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U31^11, mark1]

Status:
U31^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(153) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(154) PisEmptyProof (EQUIVALENT transformation)

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

(155) TRUE

(156) Obligation:

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

U221(ok(X)) → U221(X)
U221(mark(X)) → U221(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(157) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(ok(X)) → U221(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U221(x1)  =  U221(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U22^11, ok1]

Status:
U22^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(158) Obligation:

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

U221(mark(X)) → U221(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(159) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(mark(X)) → U221(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U22^11

Status:
U22^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(160) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(161) PisEmptyProof (EQUIVALENT transformation)

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

(162) TRUE

(163) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(ok(X1), ok(X2)) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  U211(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U21^11
mark > U21^11

Status:
mark: multiset
U21^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

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

U211(mark(X1), X2) → U211(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(166) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(mark(X1), X2) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2)  =  U211(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U21^11, mark1]

Status:
mark1: [1]
U21^11: multiset


The following usable rules [FROCOS05] were oriented: none

(167) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(168) PisEmptyProof (EQUIVALENT transformation)

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

(169) TRUE

(170) Obligation:

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

U121(ok(X)) → U121(X)
U121(mark(X)) → U121(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(ok(X)) → U121(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121(x1)  =  U121(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[U12^11, ok1]

Status:
U12^11: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

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

U121(mark(X)) → U121(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(173) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(mark(X)) → U121(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > U12^11

Status:
U12^11: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(174) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(175) PisEmptyProof (EQUIVALENT transformation)

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

(176) TRUE

(177) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(178) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(ok(X1), ok(X2)) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  U111(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > U11^11
mark > U11^11

Status:
U11^11: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(179) Obligation:

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

U111(mark(X1), X2) → U111(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(mark(X1), X2) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  U111(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[U11^11, mark1]

Status:
U11^11: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(181) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(182) PisEmptyProof (EQUIVALENT transformation)

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

(183) TRUE

(184) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(185) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > CONS1
mark > CONS1

Status:
CONS1: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(186) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(187) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(mark(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[CONS1, mark1]

Status:
CONS1: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(188) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(189) PisEmptyProof (EQUIVALENT transformation)

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

(190) TRUE

(191) Obligation:

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

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U53(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U63(X)) → PROPER(X)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(isNatKind(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(192) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2)) → PROPER(X1)
PROPER(U52(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U91(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
cons(x1, x2)  =  cons(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
U12(x1)  =  x1
isNatList(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1)  =  x1
isNat(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  U42(x1, x2)
U43(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
U63(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
U91(x1, x2, x3, x4)  =  U91(x1, x2, x3, x4)
take(x1, x2)  =  take(x1, x2)
and(x1, x2)  =  and(x1, x2)
isNatIListKind(x1)  =  x1
isNatKind(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U112 > [PROPER1, U312, U413, U914]

Status:
U622: multiset
U522: multiset
PROPER1: [1]
U914: multiset
U312: multiset
U613: multiset
U422: multiset
U413: multiset
U112: [1,2]
and2: multiset
take2: [2,1]
U712: multiset
U212: multiset
cons2: multiset
U513: multiset


The following usable rules [FROCOS05] were oriented: none

(193) Obligation:

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

PROPER(U12(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(isNatKind(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(194) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U12(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U12(x1)  =  U12(x1)
isNatList(x1)  =  x1
U22(x1)  =  x1
isNat(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
U53(x1)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
isNatIListKind(x1)  =  x1
isNatKind(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, U121]

Status:
PROPER1: multiset
U121: multiset


The following usable rules [FROCOS05] were oriented: none

(195) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(isNatKind(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(196) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U22(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
U22(x1)  =  U22(x1)
isNat(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
U53(x1)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
isNatIListKind(x1)  =  x1
isNatKind(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U221 > PROPER1

Status:
PROPER1: multiset
U221: [1]


The following usable rules [FROCOS05] were oriented: none

(197) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(isNatKind(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNat(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNat(x1)  =  isNat(x1)
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
U53(x1)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
isNatIListKind(x1)  =  x1
isNatKind(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isNat1]

Status:
PROPER1: multiset
isNat1: multiset


The following usable rules [FROCOS05] were oriented: none

(199) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(isNatKind(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(200) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNatKind(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
U53(x1)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
isNatIListKind(x1)  =  x1
isNatKind(x1)  =  isNatKind(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isNatKind1]

Status:
PROPER1: multiset
isNatKind1: multiset


The following usable rules [FROCOS05] were oriented: none

(201) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isNatIListKind(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(202) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNatList(X)) → PROPER(X)
PROPER(isNatIListKind(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
U53(x1)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
isNatIListKind(x1)  =  isNatIListKind(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
isNatList1: multiset
isNatIListKind1: multiset


The following usable rules [FROCOS05] were oriented: none

(203) Obligation:

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

PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(204) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U53(X)) → PROPER(X)
PROPER(U63(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
U53(x1)  =  U53(x1)
U63(x1)  =  U63(x1)
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U631 > [PROPER1, U531]

Status:
PROPER1: multiset
U531: multiset
U631: multiset


The following usable rules [FROCOS05] were oriented: none

(205) Obligation:

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

PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(206) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U81(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U32(x1)  =  x1
U43(x1)  =  x1
isNatIList(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  U81(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
PROPER1: multiset
U811: multiset


The following usable rules [FROCOS05] were oriented: none

(207) Obligation:

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

PROPER(U32(X)) → PROPER(X)
PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(208) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U32(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U32(x1)  =  U32(x1)
U43(x1)  =  x1
isNatIList(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, U321]

Status:
PROPER1: multiset
U321: multiset


The following usable rules [FROCOS05] were oriented: none

(209) Obligation:

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

PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(210) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(s(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U43(x1)  =  x1
isNatIList(x1)  =  x1
s(x1)  =  s(x1)
length(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, s1]

Status:
PROPER1: multiset
s1: multiset


The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

PROPER(U43(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(212) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U43(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U43(x1)  =  U43(x1)
isNatIList(x1)  =  x1
length(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, U431]

Status:
PROPER1: multiset
U431: multiset


The following usable rules [FROCOS05] were oriented: none

(213) Obligation:

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

PROPER(isNatIList(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(214) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNatIList(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatIList(x1)  =  isNatIList(x1)
length(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, isNatIList1]

Status:
PROPER1: multiset
isNatIList1: multiset


The following usable rules [FROCOS05] were oriented: none

(215) Obligation:

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

PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(216) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(length(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
length1 > PROPER1

Status:
PROPER1: multiset
length1: multiset


The following usable rules [FROCOS05] were oriented: none

(217) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(218) PisEmptyProof (EQUIVALENT transformation)

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

(219) TRUE

(220) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U91(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1, x2)  =  x1
cons(x1, x2)  =  x1
U12(x1)  =  x1
U21(x1, x2)  =  x1
U22(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  x1
U43(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2)
U52(x1, x2)  =  x1
U53(x1)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U71(x1, x2)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
U91(x1, x2, x3, x4)  =  U91(x1, x2, x3, x4)
take(x1, x2)  =  take(x1, x2)
and(x1, x2)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U512, U914, take2]

Status:
U914: multiset
U413: multiset
take2: multiset
ACTIVE1: [1]
U512: multiset


The following usable rules [FROCOS05] were oriented: none

(222) Obligation:

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

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(X)
ACTIVE(and(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(223) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U53(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1, x2)  =  U11(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U12(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  x1
U53(x1)  =  U53(x1)
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
U71(x1, x2)  =  U71(x1)
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  x1
and(x1, x2)  =  and(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U711]

Status:
cons2: multiset
U531: multiset
and2: multiset
U112: multiset
U711: multiset
U212: multiset
ACTIVE1: [1]


The following usable rules [FROCOS05] were oriented: none

(224) Obligation:

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

ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(225) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U81(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U12(x1)  =  x1
U22(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
U81(x1)  =  U81(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U811: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(226) Obligation:

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

ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(227) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(length(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U12(x1)  =  x1
U22(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
s(x1)  =  x1
length(x1)  =  length(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, length1]

Status:
length1: [1]
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(228) Obligation:

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

ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(229) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U12(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U12(x1)  =  U12(x1)
U22(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
s(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
trivial

Status:
U121: [1]
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(230) Obligation:

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

ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(231) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U22(x1)  =  U22(x1)
U31(x1, x2)  =  x1
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1
s(x1)  =  s(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
s1 > ACTIVE1

Status:
U221: multiset
s1: [1]
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(232) Obligation:

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

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(233) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U31(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U311]

Status:
U311: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(234) Obligation:

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

ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U52(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(235) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U52(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U32(x1)  =  x1
U42(x1, x2)  =  x1
U43(x1)  =  x1
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U522]

Status:
U522: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(236) Obligation:

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

ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U63(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(237) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U32(x1)  =  U32(x1)
U42(x1, x2)  =  U42(x1, x2)
U43(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U422 > [ACTIVE1, U321, U622]
U613 > [ACTIVE1, U321, U622]

Status:
U622: multiset
U422: [2,1]
U613: multiset
U321: [1]
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(238) Obligation:

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

ACTIVE(U43(X)) → ACTIVE(X)
ACTIVE(U63(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(239) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U43(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U43(x1)  =  U43(x1)
U63(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, U431]

Status:
U431: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(240) Obligation:

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

ACTIVE(U63(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(241) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U63(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
U631 > ACTIVE1

Status:
U631: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(242) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(243) PisEmptyProof (EQUIVALENT transformation)

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

(244) TRUE

(245) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1, V2)) → mark(U62(isNat(V1), V2))
active(U62(tt, V2)) → mark(U63(isNatIList(V2)))
active(U63(tt)) → mark(tt)
active(U71(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) → mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0, IL)) → mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) → mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(U63(X)) → U63(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
U63(mark(X)) → mark(U63(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(U63(X)) → U63(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U63(ok(X)) → ok(U63(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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