Problem Transformed CSR 04 OvConsOS complete C

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 OvConsOS complete C

stdout:

MAYBE

Problem:
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))

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 OvConsOS complete C

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 OvConsOS complete C

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    innermost runtime-complexity with respect to
  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))}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 OvConsOS complete C

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 OvConsOS complete C

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    runtime-complexity with respect to
  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))}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds