Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
  ↳6 QDP

(0) Obligation:

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

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ZEROS → CONS(0, n__zeros)
ZEROS → 01
U111(tt, V1) → U121(isNatList(activate(V1)))
U111(tt, V1) → ISNATLIST(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → U221(isNat(activate(V1)))
U211(tt, V1) → ISNAT(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U311(tt, V) → U321(isNatList(activate(V)))
U311(tt, V) → ISNATLIST(activate(V))
U311(tt, V) → ACTIVATE(V)
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U411(tt, V1, V2) → ISNAT(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V2) → U431(isNatIList(activate(V2)))
U421(tt, V2) → ISNATILIST(activate(V2))
U421(tt, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U511(tt, V1, V2) → ISNAT(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V1)
U511(tt, V1, V2) → ACTIVATE(V2)
U521(tt, V2) → U531(isNatList(activate(V2)))
U521(tt, V2) → ISNATLIST(activate(V2))
U521(tt, V2) → ACTIVATE(V2)
U611(tt, L) → S(length(activate(L)))
U611(tt, L) → LENGTH(activate(L))
U611(tt, L) → ACTIVATE(L)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(nil) → 01
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__0) → 01
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 27 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U611(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U521(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U521(tt, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
U511(tt, V1, V2) → ACTIVATE(V1)
U511(tt, V1, V2) → ACTIVATE(V2)
U111(tt, V1) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
U611(tt, L) → ACTIVATE(L)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

(6) Obligation:

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

U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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